T  A 


UC-NRLF 


753 


COORDINATES  OF 
ELEMENTARY  SURVEYING 


J.  C.  L.  FISH 


' 


LIBRARY 

OF  THE 

UNIVERSITY  OF  CALIFORNIA. 


Class 


COORDINATES  OF  ELEMENTARY  SURVEYING 


By  J.  C.  L.  FISH 


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COORDINATES  OF 
ELEMENTARY    SURVEYING 


BY 


J.  C.  L.     lSH,  M.  AM.  soc.  C.  E. 


Member  American  Railway  Engineering  and  Maintenance  of  Way  Association; 

some  time  Division  Engineer  Lake  Shore  and  Michigan  Southern 

Railway;  Professor  of  Railroad  Engineering  in 

Leland  Stanford  Jr.  University 


STANFORD  UNIVERSITY,  CALIFORNIA 
THE   AUTHOR 
1909 


COPYRIGHT,  1909,  BY 
JOHN  CHARLES  LOUNSBURY  FISH 


STANFORD  UNIVERSITY 
PRESS 


UNIVERSITY 

Of 


PREFACE 


In  my  first  eight  years  of  teaching"  surveying  in  the  custom- 
ary way,  stress  was  laid  on  those  fundamental  principles  of  the 
subject  which  are  independent  of  the  circumstances  of  surveying, 
namely  personnel,  equipment,  nature  of  the  survey  and  of  the 
ground.  Nevertheless  it  was  to  be  observed  that  on  subsequently 
taking  up  the  study  of  railroad  surveying  too  many  of  the  stu- 
dents were  deficient  in  grasp  of  fundamentals.  Six  years  ago 
the  experiment  was  tried  of  devoting  a  part  of  the  lectures  of 
the  course  in  surveying  to  the  fundamentals  exclusively,  and  the 
following  year  the  students  were  supplied  with  a  mimeograph 
text  covering  this  ground  and  were  required  to  work  many  prob- 
lems in  connection  therewith.  The  result  of  the  experiment  was 
gratifying. 

During  the  following  three  and  a  half  years  I  had  occasion 
to  observe  critically  the  work  of  several  surveying  parties  em- 
ployed on  heavy  railroad  construction  and  was  struck  by  the 
fact  that  nearly  all  of  the  cases  of  defective  surveying  were  due 
to  lack  of  grasp  of  the  coordinates  of  surveying  rather  than  to 
want  of  skill  in  the  use  of  implements.  The  lack  of  familiarity 
with  fundamentals  was  most  often  exhibited  in  failure  to  devise 
a  scheme  of  check  measurements  which  should  be  independent 
of  the  original  measurements. 

It  has  been  observed  not  infrequently  that  the  young  sur- 
veyor looks  upon  the  different  branches  of  surveying,  e.  g.,  land, 
city,  mine  and  hydrographic  surveying,  as  involving  different 
fundamental  principles  of  surveying,  and  this  is  because  in  his 
training  the  varying  circumstances  of  surveying  have  been  per- 
mitted to  overshadow  the  unvarying  principles. 

The  foregoing  facts  supply  the  reason  for  making  much  of 
and  formally  presenting  the  subject  of  this  book. 


191650 


VI  PREFACE 

The  general  absence  of  explanations  is  due  to  design :  to 
make  the  book  brief,  advantage  has  been  taken  of  the  fact  that 
in  this  institution  the  study  of  coordinate  geometry  precedes  the 
study  of  surveying.  It  is  assumed  that  the  teacher  who  uses 
this  drill  book  will  enliven  it  by  illustrations  taken  as  far  as 
possible  from  the  experiences-  of  his  students,  and  require  from 
them  accurate  and  neatly  arranged  solutions  of  numerous  prob- 
lems. Ten  to  fourteen  lectures  may  be  profitably  devoted  to 
illustrating  the  text,  and  the  student  may  well  give  upwards  of 
twenty  hours  to  the  solution  of  original  problems. 

The  Tables  relating  to  interchange  of  bearings,  azimuth  and 
deflections  are  intended  to  be  used  only  in  checking  computation. 

Lest  the  forbidding  mathematical  aspect  of  some  of  the 
pages  should  discourage  the  reader  on  first  opening  the  book, 
it  ought  to  be  said  here  that  the  subscript  is  used  to  gain  brevity 
without  loss  of  precision  of  statement,  and  does  not  indicate  the 
presence  of  "higher  mathematics." 


CONTENTS 


PART  I.— INTRODUCTION 
CHAPTER  I. —  GENERAL  STATEMENTS 

ARTICLE  PAGE 

2.  Definitions  .      .      .      .      .      .      ...      .      .            ...  3 

3.  Survey  of  Straight  Line      .      .      <;      .      .      .      .      .  .  4 

4.  Circle.        .      .      .      . .4 

5.  Irregular  Curves       .      .      .      .      ;      .      .  .  4 

6.  of  a  Plane      .      .      .      ;      .      .      .      ;.     .  .  4 

7.  Irregular  Surfaces    .              4 

CHAPTER  II. — -SYSTEMS  OF  COORDINATES 

9.  Systems  of  Coordinates  in   Horizontal   Surveying   .  .  5 

10.  Vertical                             .  .  6 

11.  Space                                 .  .  6 

CHAPTER   III. —  NOTATION      • 

12.  Symbols ,  .  # 

13.  Greek  Alphabet          .      * ;      .  .  7 


PART  II.—  HORIZONTAL  SURVEYING 

CHAPTER  IV. — -RECTANGULAR  COORDINATES 

14.  Rectangular  Coordinates     _.      .      .      .      .      .      .      .  .       8 

15.  General  and  Local  Rectangular  Coordinates        .      .  .        9 

16.  .r- Difference  and  ^'-Difference          .      .      .      /     .      .  .      10 

17.  Equation  of  a  Straight  Line     .      .      .      .      .      .  .    .  .      10 

18.  Intersection  of  Two  Straight  Lines 11 

19.  Area  of  Closed  Figure         12 

20.  From   General  to   Local   Coordinates  14 


viii  CONTENTS 

ARTICLE  PAGK 

21.               Local  to  General   Coordinates:   Single   Point      .  14 

Series  of   Points  14 

23.  To  Make  Coordinates  All  Positive     .    ».      .      .      .      .  15 

24.  From   Local   Rectangular   to   Local    Polar   Coordinates  15 

25.  General                              Polar   Coordinates      .      .  17 

CHAPTER  V. —  POLAR  COORDINATES 

26.  Polar  Coordinates .      .      .      .      .  18 

27.  Direction  and  Distance          . 18 

28.  Azimuth                                     18 

29.  Bearing  .......      .      .19 

30.  Deflection  .      .    ' .      .20 

31.  Angle   Between   Two  Lines:   Azimuths     .     -,      .      .      .  21 

32.  "                                                Bearings 21 

33.  Distance    Between   Two    Points       .      .      ...      .      .  22 

34.  Area  of   Polygon 23 

35.  To  Change  Zero  Direction :  Azimuth 24 

36.  "                          Bearings 25 

37.  From  Azimuth  to   Bearing 27 

38.  "                 "            Deflection          .......  27 

39.  "       Bearing   to   Azimuth 28 

40.  "                 "            Deflection 28 

41.  "       Deflection   to   Azimuth 29 

42.  "                ."               Bearing 30 

43.  To   Change   Pole   Without   Changing  Zero   Direction : 

Azimuths _  *      .      .32 

44.  From  Local  Polar  to  Local  Rectangular  Coordinates : 

Azimuths 32 

45.  From  Local  Polar  to  Local  Rectangular  Coordinates : 

Bearings 33 

46.  From  Local  Polar  to  Local  Rectangular  Coordinates : 

Deflections         .      .      .      .      .      .      ...      .      .      .  35 

47.  From   Polar   Coordinates   to   Biangular   Coordinates      .  35 

CHAPTER  VI. —  BIANGULAR  COORDINATES 

48.  Biangular   Coordinates    .      .      ,      .      .      .      .      .     -.      •  36 

49.  Distance  Between  Two  Points        .      .      .      ,      .      .      .  36 

50.  Area  of  the  Triangle     .      .      .     ,.     .      .      ...      •  38 


CONTENTS  IX 

ARTICLE  PAGE 

51.  From  Biangular.  to  Polar   Coordinates:   Azimuth   and 

Distance       .;    ,      .      .      .      .      .      .      .      .      .     ..;.    .  38 

52.  From   Biangular  to    Polar   Coordinates :    Bearing   and 

Distance       ,      ..      .      .      .      .      .      ....      .      .  39 

53.  From  Biangular  to  Biradial  Coordinates :   Single  Point  40 

54.  "'  Series  of 

Points 40 

55.  From  Biangular  to  Rectangular  and  Other  Coordinates  43 

CHAPTER  VII. — •  BIRADIAL   COORDINATES 

56.  Biradial    Coordinates 44 

57.  Area  of  the  Triangle 44 

58.  From  Biradial  to  Biangular  Coordinates        ....  44 

59.  "                            Polar                                      ....  45 

60.  "                             Rectangular   and    Other    Coordinates  46 

CHAPTER  VIII. —  BIPOLAR   COORDINATES 

61.  Bipolar    Coordinates 47 

62.  From   Bipolar  to   Polar  Coordinates          47 

63.  Rectangular    and    Other    Coordinates  47 

CHAPTER  IX. —  TRIPOLAR  COORDINATES 

64.  Tripolar   Coordinates .49 

65.  From  Tripolar  to   Polar   Coordinates 49 

66.  "                               Rectangular  and  Other  Coordinates  51 


PART   III.— VERTICAL  SURVEYING 

CHAPTER  X. —  RECTANGULAR  COORDINATES 

67.  Rectangular    Coordinates     ^      .      .      .      .      ...  .      52 

68.  .From  General  to  Local  Rectangular  Coordinates    .  .      53 

69.  Local  to  General                                               ^  .  .      53 

70.  '*             "     Rectangular    to    Polar                             .  .      54 

71.  "       General           "                     "                                 .  .      54 

72.  Local                                              Rectangular 
Coordinates  54 


X  CONTENTS 

CHAPTER  XI. —  POLAR  COORDINATES 

ARTICLE 

73.  Polar   Coordinates "...      56 

74.  From   Polar  to  Local  Rectangular   Coordinates        .      .      56 

75.  "  Polar-Rectangular  .      .      56 

CHAPTER  XII. — •  POLAR-RECTANGULAR  COORDINATES 

76.  Polar-Rectangular    Coordinates       .      .      .      .      .      .      .      58 

77.  From  Polar-Rectangular  to  Local  Rectangular 

Coordinates 58 

78.  From   Polar  Rectangular  to   Polar  Coordinates        .      .      58 


PART  IV.— SPACE  SURVEYING 

CHAPTER   XIII. —  RECTANGULAR   COORDINATES 

79.  Rectangular    Coordinates 60 

80.  From   General   to   Local   Rectangular   Coordinates         .     61 

81.  "       Local   to   General 

Single   Point  61 

82.  From   Local  to  General  Rectangular  Coordinates: 

Series   of   Points 62 

83.  From  General  Rectangular  to   Polar   Coordinates   .      .     63 

84.  "  -Rectangular 

Coordinates 64 

CHAPTER  XIV. —  POLAR   COORDINATES 

85.  Polar   Coordinates 65 

86.  From   Polar  to    Polar-Rectangular   Coordinates        .      .     65 

87.  Local  Rectangular  .      .     65 

88.  "  General  Rectangular  .      .     66 

CHAPTER  XV. —  POLAR-RECTANGULAR  COORDINATES 

89.  Polar-Rectangular    Coordinates       ........     67 

90.  From  Polar-Rectangular  to  Local  Rectangular 

Coordinates       .      ,      .,     .      .      ,      .      .      .      ...     67 

91.  From    Polar-Rectangular   to    General    Rectangular 

Coordinates   , .  ,,     .      .      .      .      .      .      «      ....     67 

92.  From   Polar-Rectangular   to    Polar   Coordinates        .      .     67 


PART   I.— INTRODUCTION. 


CHAPTER   I. 

GENERAL  STATEMENTS. 

1.  For  the  purpose  of  study  the  subject  of  Elementary  Survey- 
ing may  be  divided  into  the  following  parts  :    ( i )  Coordinates  of 
surveying,  which  are  the  "magnitudes  which  serve  to  fix  the  position 
of  a  point, —  in  space,  on  a  surface,  or  on  a  line."      (2)  Surveying 
instruments,  by  means  of  which  the  magnitudes  are  measured.     (3) 
Errors  of  measurement.      (4)    Methods  of  surveying,  which  deal 
with  the  practical  application  of  systems  of  coordinates,  either  singly 
or  in  combination,  through  the  use  of  various  surveying  instruments. 
(5)    Computing  and   mapping,    by  which   the   measurements   are 
combined  and  the  results  presented  for  convenient  use. 

Coordinates  of  surveying,  unlike  the  other  four  topics,  have  to 
do  only  with  mathematical  laws.  A  thorough  grasp  of  the  coordi- 
nates of  surveying  is  absolutely  necessary  to  the  intelligent  planning 
of  a  survey,  whether  it  involves  few  or  many  points. 

This  book  deals  only  with  the  systems  of  coordinates  used  in 
surveying,  and  the  following  definitions  are  accordingly  restricted. 

2.  Definitions :    Surveying. —  Surveying    consists    in    making 
measurements  of  angles  or  of  distances  or  of  both,  with  the  object 
(i)  of  finding  the  mathematical  relations  which  exist  between  given 
points  of  the  earth's  surface  ;  or  (2)  of  marking  on  the  earth's  sur- 
face points  which  bear  given  mathematical  relations  to  given  points 
of  the  earth's  surface. 

A  survey  made  with  the  second  object,  just  stated,  is  called  a 
location  survey  or  simply  a  "location";  and  the  marked  points  are 
said  to  be  "located." 

In  Horizontal  Surveying,  all  the  points  considered  lie  in  the  same 
horizontal  plane,  and  are  either  actual  points  or  the  horizontal 
projections  of  actual  points. 

In  Vertical  Surveying,  all  the  points  considered  at  one  time  lie 
in  the  same  vertical  plane. 


4  COORDINATES    OF     ELEMENTARY    SURVEYING      • 

In  Space  Surveying,  the  points  are  considered  in  their  actual 
positions  in  space. 

3.  Survey  of   Straight  Line. —  A  limited  straight  line  is  deter- 
mined by  its  extremities.     Therefore,  to  survey  a  system  of  straight 
lines  it  is  sufficient  to  determine  the  position  of  the  two  extremities 
of  each  line.     When  two  lines  have  a  common  extremity,  the  two 
lines   are   determined  by   the   common   point   and   the   two  other 
extremities.     In  general,  a  broken  line  is  determined  by  a  number 
of  points  one  greater  than  the  number  of  segments  of  the  broken  line. 
In  the  case  of  a  polygon,  as  the  boundary  of  a  field,  two  adjacent 
lines  have  a  common  extremity  and  the  n  lines  of  the  figure  may  be 
surveyed  by  determining  the  positions  of  the  n  double  extremities. 

4.  Survey  of  a  Circle. —  To  survey  a  circle  it  is   sufficient   to 
determine  the  radius  and  the  position  of  the  center  of  the  circle,  or 
to  determine  any  three  points  of  the  circumference. 

5.  Survey  of  Irregular  Curves. —  When  it  is  required  to  survey 
an  irregular  curve  (whether  it  lies  in  a  horizontal  or  a  vertical  plane, 
or  is  a  non-planar  curve)  the  surveyor  determines  the  positions  of  a 
number  of  points  on  the  line.     The  distribution  and  number  of 
points  determined   in  any  survey  depend  upon  the  object  of  the 
survey.    The  surveyor  chooses  the  points  so  that  for  the  purpose  for 
which  the  survey  is  required  the  resulting  segments  of  the  curve  may 
be  considered  as  straight  lines,  or  the  smooth  curve  drawn  through 
the  points  may  be  considered  as  the  true  curve.     Theoretically  the 
accuracy  of  the  survey  of  the  curve  depends  upon  the  number  of 
points   determined,    other   things   being   equal ;    but   the  labor   of 
surveying  increases  with  the  number  of  points  determined. 

6.  Survey  of  a  Plane.— To  survey  a  plane  it  is  sufficient  to 
determine  the  positions  of  three  points  (which  do  not  lie  on  the  same 
straight  line)  of  the  plane.   (We  are  not  here  considering  the  survey 
of  a  limited  plane  area.  ) 

The  ground  surface  is  not  plane,  but  limited  areas  sometimes 
are  (or  are  made  to  be)  approximately  plane,  and  are  often  con- 
sidered to  be  plane. 

7.  Survey  of  Irregular  Surfaces. —  To  survey  an  irregular  sur- 
face the  surveyor  determines  the  positions  of  a  number  of  points  of 
the  surface.     The  points  are  selected  with  regard  to  number  and 
distribution,  so  that,  for  the  purposes  of  the  survey,  the  broken  plane 
surface   determined   by   the   points   or   the  smooth  surface  drawn 
through  the  points,  may  be  taken  to  represent  the  actual  surface. 


SYSTEMS     OF    COORDINATES 


CHAPTER   II. 
SYSTEMS  OF  COORDINATES. 

8.  The  distances  or  angles,  or  distances  and  angles,  which  are 
measured  for  the  purpose  of  determining  the  position  of  any  point 
with  respect  to  any  given  points  or  lines,  are  called  coordinates  of 
the  point.      In  Elementary  Coordinate  Geometry  are  discussed  two 
systems  of  coordinates,  —  Rectangular  and  Polar.    These  and  several 
other   systems  which  in  this  book  are  given  arbitrary  names,   are 
used  in  surveying.      These  arbitrary   names   are  not  used  in  the 
practice  of  surveying,  and  are  introduced  here  simply  for  convenience 
in  discussing  systems  of  coordinates. 

9.  Systems  of  Coordinates  in  Horizontal  Surveying : 

Y 


0. 

FIG.  i 
Rectangular  coordinates 


FIG.  2 
Polar  coordinates 


a) 


P.  Pa 

FIG.  3 
Biangular  coordinates  * 


P,  Pa 

FIG.  4 
Biradial  coordinates  f 


FIG.  5 
Bipolar  coordinates  \ 


Pa 

FIG.  6 
Tripolar  coordinates 


*  Referred  to  by  Pence  and  Ketchum  in  Surveying  Manual  as  "angular 
intersection." 

t "  Focal  coordinates  or  tie  lines  "  of  Surveying  Manual. 
{  "  Modified  polar  coordinates  "  of  Surveying  Manual. 
\  "  Resection  "  of  Surveying  Manual. 


6  COORDINATES    OF    ELEMENTARY    SURVEYING 

10.  Systems  of  Coordinates  in  Vertical  Surveying  : 


Datum 


FIG.  7 
Rectangular  coordinates 


FIG.  9 
Polar-rectangular  coordinates 

1 1 .  Systems  of  Coordinates  in  Space  Surveying  -. 


P 

t 

Iz. 


-^ X 

FIG.  10 

Rectangular  coordinates 
(JTand  Y  are  horizontal) 


FIG.  8 
Polar  coordinates 


FIG. ii 

Polar  coordinates 
X  and  Y  are  horizontal ) 


Frc.  12 
Polar-rectangular  coordinates 


NOTATION 


CHAPTER    III. 
NOTATION. 

12.  Symbols. —  The  following  symbols  are  typical  of  the  uniform 
notation  used  throughout  this  book. 
Plt    P,,    .    .  .  Pky    points. 
;F,      abscissa 


of  the  point  /\  referred  to  a  general  origin. 


of  the  point  P.2  referred  to  the  origin  Pl . 


j/!  ordinate 

Zi  elevation 

jtr,.2  abscissa 

yr.2  ordinate 

zrt  elevation 

A  x . ,  X- 

AjY  a  }'~  difference  of  the  points  Pl  and  P2  = 

A^.JJ  s- 

ar2  azimuth 

/a,.2  true  azimuth 

;«ar  2  magnetic  azimuth 

j3r  2  bearing 


*-2       *\  i 

r2— 7i; 

s*.,  —  #, . 


true  bearing  of  the  line 

magnetic  bearing 

deflection  angle  to  the 


of  the  line  P1P,. 


right  ; 
left. 


dr^      (horizontal)  distance 
sdr  .2     slope-distance 


between  the  two  points  Pl  and 


°"r-2 

D 
&i 


slope-angle  of  the  line  Pl  P.2 . 
datum-line,  or  plane,  through 


Y, 


general  origin  ; 
origin  Pl . 


rectangular  axes  through 


general  origin  ; 


origin 
the  (horizontal)  angle  Pl  P^  P3 . 


13.  Greek  Lower-Case  Letters. 
a  alpha  f  zeta 

/3  beta  77  eta 

7  gamma  6  theta 

&  delta  (lower  case)        L    iota 
A  delta  (capital)  K  kappa 

e    epsilon  X  lambda 

a  mil 


v  nu 

r  tau 

f  xi 

v  upsilon 

o  omicron 

<f>  phi 

TT  pi 

%chi 

/o  rho 

i/rpsi 

a  sigma 

oj  omega 

PART  II.— HORIZONTAL  SURVEYING. 


CHAPTER   IV. 
RECTANGULAR  COORDINATES. 

14.  Rectangular  Coordinates. —  In  Fig.  13  is  shown  a  point  Pn 
referred  to  the  rectangular  coordinate  axes  Xk  and  Yk  which  in- 
tersect at  the  origin  Pk ,  by  the  rectangular  coordinates  xk.n  and 
yk.n .  xk.n  is  the  abscissa,  and  yk.n  is  the  ordinate,  of  Pn . 


Quad.ET 


fX 


Quad.  I 


Xoeg 
ypos. 


X  neg. 
yneg. 


Xpos.    | 
ypos.    j, 


Xpos. 


Quad.IH        _ 


FIG.  13 


Quad.H 


F? 


-X 


FIG.  14 


We  number  the  quadrants,  I,  II,  III,  IV,  as  shown  in  Fig.  13, 
beginning  with  that  quadrant  between  the  two  arrow-heads,  and 
counting  in  clockwise  direction.  It  is  customary  to  consider  the 
coordinates  positive  or  negative  according  to  the  following  table  : 

SIGN  OF  X  AND  OF  y  FOR  EACH  QUADRANT. 


Quad.  I 

Quad.  II  ' 

Quad.  Ill 

Quad.  IV 

X 

y 

+ 

-f 

_j_ 

— 

+ 

RECTANGULAR    COORDINATES  9 

Example.  —  Fig.  14  represents  a  piece  of  "coordinate  paper" 
upon  which  have  been  marked  the  two  axes  JT  and  Y  with  origin 
P0 ,  and  four  points  /\ ,  P2 ,  P3 ,  Pt.  One  side  of  a  small  square  is 
taken  as  unity  in  measuring  the  coordinates  of  the  points.  Using 
symbols  we  express  the  coordinates  of  the  four  points  thus: 


Required  to  rewrite  these  expressions  substituting  for  each  x  and 
each  y  its  numerical  value  with 
proper  sign. 

The  result  is  : 


COORDINATES  OF  POINTS. 
ORIGIN  AT  P0. 


•^1(3,  5),  P*(—  i,  —3), 

P2(4,  -4),  />,(— 3,  2), 

in  which  those  quantities  which 
are  not  preceded  by  the  sign  ( — ) 
are  understood  to  be  positive. 
For  convenience  we  may  express  the  result  as  shown  in  this  table. 
15.  General  and  Local  Rectangular  Coordinates. —  If  all  the 
points  of  a  survey  are  referred  to  the  same  origin  and  axes  — 


Point 

x 

y 

P 

3 

5 

P. 

4 

—4 

P-i, 

—  i 

—3 

P± 

—3 

2 

The 


origin 

axes 

coordinates 


will  be  called  general 


origin ; 
axes ; 
coordinates. 


E.  g. ,  the  general  coordinates  of  a  point  Pn  are  xn  and  yn ,  and 
when  xoryis  written  with  a  single  subscript  we  understand  that  it 
refers  to  the  general  origin  and  axes. 

If  the  points  of  a  survey  are  divided  into  groups  and  each  group 
is  referred  to  a  separate  origin,  then  for  each  group 


The 


origin 

axes 

coordinates 


will  be  called  local 


origin  ; 

axes; 

coordinates. 


If  the  local  origin  is  Ptt  the  local  axes  are  Xk  and  Yk,  and  the 
local  coordinates  of  a  point  Pn  (referred  to  P^)  are  xk.n ,  yk.n ;  and 
when  x  or  y  is  written  with  a  double  subscript  we  understand  that 
the  first  part  of  the  subscript  refers  to  the  local  origin,  and  the  sec- 
ond part  refers  to  the  point  determined. 


10  COORDINATES    OF     ELEMENTARY    SURVEYING 

16.  jr-Difference  and  y-Difference.—  If  two  points  Pk  and  PH  are 
referred  to  the  same  origin  so  that  we  have 


then  of  the  points  Pk  and  Pu  the  ^"difference  is  {*?*""  Zf"  ~  **' 

JT~  ^y  k-n          J  n  y  k  > 

in  which  the  sign  of  the  result  is  to  be  regarded. 

The  ^"difference  of  a  limited  straight  line  is  the  ab^lssa  ofthe 
y-  ordinate 

terminal  point  diminished  by  the  Qr  jj^^  °f  tne  initial  point  of  the 
line  ;  e.  g., 

%**'*  II  **  ~  **  II  ^'difference  of  line  Pk  Pn  ,  the  sign  of  the  re- 
sult to  be  regarded  ;  while 

&*„.*  -=  xk--xn  =  ^-difference  of  the  line  PH  Pky  the  sign  of  the 

•*y»'jt  —  y  *    **  —  y~ 

result  to  be  regarded. 

Terms  equivalent  to  ^-difference  and  ^-difference,  to  be  found  in 
books  on  surveying,  are  given  in  the  following  table: 

EQUIVALENT  TERMS. 


x-  Difference 

^/-Difference 

Longitude  difference 
Longitude 
Departure 
J  Easting 
(  Westing 

Latitude  difference 
Latitude 
Latitude 
|  Northing 
(  Southing 

17.  Equation  of  a  Straight  Line. —  The  equation  of  a  straight 
line  which  passes  through  the  two  points  Pk  (xk,  y^  and  Pn  (xtlt  y,^) 
is  commonly  written 

y^ax-^b, 

in  which  a  =  yn  —  yk  jxH  —  xk 

and  b  =  yA  —  axk . 

Example.  — Write  the  equation  of  the  line  which  passes  through 
points  P,(2,  5)  and  P.2 (6,  3). 

We  use  the  following  algorithm  in  solving. 


RECTANGULAR    COORDINATES 


11 


ALGORITHM  :    EQUATION  OF  STRAIGHT  LINE. 


I 

Line  i"2 

2 

Xk 

2. 

3 

x* 

6. 

4 

y* 

5- 

5 

}'n 

3- 

6 

y*  —y* 

2. 

7 

*n  —  *k 

4- 

8 

log  (v,t—  y*) 

o  .  30  i  n 

9 

log  (xn  —  XA) 

0.602 

10 

log  a 

9  .  69911 

ii 

a 

—0.5 

12 

log  4-,, 

9.301 

13 

log  axk 

o.ooon 

14 

axk 

—  I  .0 

15 

b  (=>*—£*•*) 

6. 

16 

Eq.  :  y  —  ax-\-b 

y  =  —  o  .  *x  +  6 

A 

B 

18.  Intersection  of  Two  Straight  Lines. —  Given  the  equation  of 
each  of  two  straight  lines,  /and  /',  to  find  the  coordinates  xt ,  jz  of  their 
point  of  intersection,  Pt . 

The  equations  are : 

for  line  /,  y  =  ax  -f  b  .         •         •          (0 

and  for  line  /'  y  =  a'x  -\~  b'  ...          (2) 

Putting  (ax  -\-  b}  for  y  in  (2)  we  obtain 

ax  -\-  b  =  a 'x  -\-  b' 
or,  (a  —  a')x=  (b'  --  b). 

We  can,  therefore,  say  :  the  abscissa  of  the  point  of  intersection  is 

*,=  (b'  •-  b}  I  (a  —  «'), 
and  the  ordinate  of  the  point  of  intersection  is 

y.  =  aXi  -r  b, 
and  the  check  equation  is 

yt  =  a'Xi  +  b'. 

Example.  —  Find  the  coordinates  of  the  point  of  intersection  P& 
of  the  two  lines  Pl  P^  and  P3  Pi  whose  equations  are  respectively 

y  =  0.5*+  4 
y  =  zx  -f-  7- 


12  COORDINATES    OF     ELEMENTARY    SURVEYING 

ALGORITHM  :    INTERSECTION  OF  Two  LINES. 


I 

/>.   =  (P.P.,,  P3  PJ 

2 

Eq.  y  =  ax  -f  b 

y  =  0.5*4-4 

3 

Eq.  y  =  a'x-\-  b' 

^/  =  2*4-  7 

4 

a 

0.5 

5 

a' 

2. 

6 

b 

4- 

7 

b' 

7- 

8 

b'  —  b 

3- 

9 

a  —  a 

—  1.5 

10 

log.  (b'—b) 

0.477 

ii 

log.  (a—  a') 

o.  I76n* 

12 

log.  Xi  =  diff.  log. 

0.30  in 

13 

log.  a 

9.699 

14 

log.  a 

0.301 

15 

log.  axi 

o.  ooon 

16 

log.  a'Xi 

o.  6o2n 

17 

axi 

—  i.o 

18 

a'Xi 

—4.0 

19 

Xi 

2.0 

20 

axi  -\-  b  —  yi 

3- 

21 

a'xi-\-  b'  =  yi 

3.  (check) 

A 

B 

19.  Area  of  Closed  Figure.  —  If  Pl  P.2,  P2  P3,  .  .  .  Pn_iPn, 
PnPl  are  the  sides,  taken  in  order,  of  a  closed  plane  figure,  the  area 
of  the  figure  can  be  expressed  in  two  ways,  one  of  which  is  conven- 
ient for  slide-rule  and  the  other  for  logarithmic  computations.  For 
slide-rule  computation  we  use  this  equation  (Coord.  Geom.): 

Area  =  ^[(^  x,  +  j/a  x,  +  /3  x,  +  .  .  .  -f  yn  x,) 

-  (ji  *n  -I-  j,  ^  4-  y*  x,  +  •  -  .  -f-  }'n  *«_i)] 

in  which  x±  ,  yv  are  the  general  coordinates  (Art.  1  5)  of  point  Pl  ; 
x^  y.z  ,  of  P2  ;  etc.      Or,  in  words  : 


Multiply  the  y  of  each  point  by  the  x  of  the 


and  add  the  products,  obtaining  the  sum 
3'  The"'  Area    =  ?=!' 


*  This  "  n  "  shows  that  the  log.  is  of  a  negative  number, 
t  PI  is  the  point  following  Pn.     Pn  is  the  point  preceding  /\. 


RECTANGULAR    COORDINATES 


13 


For  logarithmic  computation  we  use  the  following  equation 
(Coord.  Geom.)  : 

Area  =  ^[j^fe  —  #„)  -f-  y£x*  —  x^  -f-  y.,(x,  —  x.t)  -f  .  .  . 

+  ^fe-^-0] 

in  which  ;r, ,  j^  are  the  general  coordinates  of  Pl ;   .r, ,  ytt  of  P.2;  etc. ; 
or,  in  words  : 

1.  Subtract  the  x  of  each  point  from  the  x  of  the  second  point 
following,*  obtaining  an  .^-difference,  A;r. 

2.  Multiply  each  A,r  by  the  y  of  the  intermediate  point. 

3.  Then,     Area  =  ^(algebraic  sum  of  products). 

Example. — Pl ,  P2,  P.A ,  Pi ,  P&  are  the  corners  of  a  field,  taken  in 
order  around  the  perimeter.  Given  Pl  (2,  i),  P2  (3,  2),  P9  (4,  — 7), 
PI  (—3,  —2),  P,  (—5,  6).  What  is  the  area  of  the  field  ? 


I 

2 

Point 

x 

J 

*-diff.  = 

log 
log^r-diff. 

Partial  area 

3 

^.-i) 

sum  log 

(+) 

(—  ) 

4 

p. 

-  5 

5 

P\ 

2 

I 

0.000 

6 

8 

0.903 

7 

0.903 

8 

8 

r> 

3 

2 

0.301 

9 

2 

0.301 

10 

0.602 

4 

ii 

r> 

4 

-7 

o.845n 

12 

—6 

0.77811 

13 

1.623 

42 

14 

P. 

—3 

—2 

o.3om 

15 

—9 

o.954n 

16 

1-255 

18 

17 

P* 

-5 

6 

0.778 

18 

5 

0.699 

19 

1-477 

30 

20 

Pl 

2 

21 

Sum 

-(-i  02 

-o 

Therefore  area  =  ^(102  —  o)  =  +5i.f 


*The  "  second  point  following 


„/-.-, 


fThe  sign  of  the  area  has  no  significance,  as  it  depends  only  on  whether 
we  take  the  points  in  clockwise  or  counter-clockwise  order. 


14  COORDINATES    OF     ELEMENTARY     SURVEYING 

20.  From  General  to  Local  Coordinates.  —  We  are  given  the  gen- 
eral coordinates  xk,  yk  of  the  point  Pk,  and  xnt  y»  of  the  point 
Pn  .  We  wish  to  know  the  local  coordinates  xk.n  ,  yk.n  of  Pn 
referred  to  the  local  axes  Xk  and  Yk  taken  through  the  local  origin, 

Pk  .     Xk  is  ||  to  X  and  Yt  is  ||  to  Y.     Or,  briefly, 

Given  Pk(x*>  yd     and     Pn(xHt  yn)  , 

To  find  Pn(xk.,t,  yk.,d- 

**•«  —  *«  —  ^  =  A*X..«  —  ^difference  of  line  PhPn. 

y»»  =y»  —yk=^  fy#»  =7- 

21  .  From  Local  to  General  Coordinates  ;  Single  Point.  —  We  are 
given  the  local  coordinates  xk.n  ,  yk.n  of  the  point  Pn  ,  and  the  gen- 
eral coordinates  xk  ,  yk  of  the  point  Pk  .  We  wish  to  know  the  gen- 
eral coordinates  xn  ,  yn  of  Pn.  X  is  ||  to  Xk  and  F  is  ||  to  Yk.  Or, 
briefly, 

Given  Pk(xk,  y^    and    Pn(xk.n,  y*.n), 

To  find  /^(^M,         - 


22.  From  Local  to  General  Coordinates  ;  Series  of  Points.  —  We 

are  given  a  series  of  points,  Pl  ,  P.2  ,  P3  ,  .  .  .  .  We  are  given  the 
general  coordinates  x^  ,  yl  ,  of  the  first  point,  /\  ,  of  the  series.  For 
each  point  after  the  first  we  have  the  local  coordinates  referred  to  the 
preceding  point  as  origin.  All  the  ^r-axes  are  ||  and  all  the  jj/-axes 
are  ||  .  We  wish  to  find  the  general  coordinates  of  all  the  points 
after  the  first.  Or,  briefly, 
Given  ^i  (X  ,  yd 


To  find  P.,  O,  »  yd 

^3<X,  yd 

The  general  coordinates  for  any  point  Pn  of  the  series  are 

xn  =~-*i  +  *i.,  -f  ^-3  +  Xy*  +..-.+  x»_rn         (i) 

y»  =y*  +^i-s  +^.3  +  ^5-4  +  •  •  •  +/»-!•«       (2) 

or,  in  words  : 


RECTANGULAR    COORDINATES 


15 


The  general 


abscissa 
ordinate 


of  any  given  point  of  the  series  is  equal 


abscissa 
ordinate 


and  the  local 


of  the  first  point 
of  all  the  points  after  the  first  point  up  to 


(3) 
(4) 


to  the  algebraic  sum  of  the  general 

abscissas 

ordinates 
and  including  the  given  point. 

If  we  are  considering  the  lines  P^P^  P*P*,  P3^\,    -   •   •   equa- 
tions (i)  and  (2)  may  be  written  in  this  form : 

in  which  A^.j  =  x^  —  xv  ===  ^-difference  of  line  PVP^ ,  etc. 
In  words  : 

abscissa 


The  general 


ordinate 


of  any  given  point  of  the  series  is  equal 


to  the  algebraic  sum  of  the   general 


and   the 


jr-diffs. 


of  the  first  point 


of  all  the  lines  between  the  first  point  and  the 


j-diffs. 
given  point. 

23.  To  Make  Coordinates  All  Positive.  —  Liability  to  error  in 
computations  involving  coordinates  is  less  where  the  coordinates  are 
all  positive  than  where  some  are  positive  and  some  negative.  The 
method  of  making  mixed  coordinates  all  positive  is  here  given. 

Given  the  general  coordinates  of  a  group  of  points,  some  of  the 
coordinates  being  negative  ;  to  find  new  general  coordinates  all 
positive.  Proceed  thus  : 

For  the  new  '  '  of  each  point  add  *  to  the  given  ;  taking  * 
positive  and  equal  to  that  power  of  10  next  larger  than  the  greatest 


negative 


This  amounts  to  referring  the  points  to  a  new  ^r-axis  taken  a 
distance  q  below  the  old  ^r-axis,.  and  a  new^-axis  a  distance  /  to 
the  left  of  the  old  jj/-axis. 

24.  From  Local  Rectangular  to  Local  Polar*  Coordinates.  — 
Given  the  rectangular  coordinates  xk.n  ,  yk.n  of  the  point  Pn  referred 
to  the  origin  Pk  .  Take  Pk  for  the  pole  and  -{-  Yk  (axis)  for  zero 
direction.  Required  the  polar  coordinates,  azimuth,  ak.n  (or 
bearing,  ft.,,),  and  the  distance  dk.n  of  Pn  referred  to  Pk  . 


*  This  article  is  to  be  read  after  Chapter  V  on  Polar  Coordinates. 


16 


COORDINATES    OF    ELEMENTARY    SURVEYING 


Or,  briefly, 

Given          Pn  (xk.n  ,  yk.^> 

and  zero  direction  =  -+-  Yk  (axis), 

Required    PH  (ak.n ,   dk.^  ; 
or  />„(&.„,    4*.M). 

From  Fig.    15  the  tangent  of  the 
direction  angle  is 

tan  ak.n  =  tan  &.„  =  xk.H\  yk.n. 


The   quadrant   in    which   the 


azimuth 


FIG.  15 
terminates    (/'.  e. ,    the 


bearing 
quadrant  in  which  Pn  lies)  is  determined  by  the  following  : 

I 

a  terminates  in       II 
quadrant          III 
IV 


If*  is 


and  y  is 


an 


d/3i 


is 


NE; 

SE; 

SW; 

NW. 


The  distance  is  dk.n  =  jr^/sin  ak.n  =  jtr^/sin  &.„. 
The  check  equation  is  <^V«  —  x<ik-n  H~  }?*•»  • 

Example. — We  are  given  the  general  rectangular  coordinates 
of  these  points :  Pl  (2,  8),  P,  (14,  —7),  P3  (—9,  —5),  P,  (—3,  4). 
Taking  the  general  origin  for  a  pole  and  -f-  F(axis)  for  a  zero  direction, 
what  are  the  azimuths  and  distances  of  the  points  ? 


I 

Point 

P, 

P* 

P. 

pt 

2 

X 

2 

14. 

—9. 

3- 

3 

y 

8 

—  7- 

—  5- 

4- 

4 

log  * 

0.301 

1.146 

o-954 

0.477 

5 

logj 

0.903 

0.845 

0.699 

0.602 

6 

log  tan  a  '  * 

9-398 

0.301 

0-255 

9-875 

7 

log  sin  a  ' 

9-384 

9-95i 

9.941 

9-779 

8 

log^ 

0.917 

i-i95 

1.013 

0.698 

9 

a' 

14.0° 

63-4° 

60.9° 

36-9° 

10 

d 

8.26 

•15-7 

10.3 

5-o 

ii 

at 

14.0° 

116.6° 

240.9° 

323-2° 

12 

log  x1 

0.602 

2.292 

1.908 

0-954 

13 

logy 

i.  806 

1.690 

1.398 

1.204 

M 

4T3 

4- 

196. 

81. 

9- 

15 

/J 

64. 

49. 

25- 

1  6. 

16 

*2  +  y 

68. 

245. 

106. 

25- 

i? 

log;/ 

1.834 

2.390 

2.026 

1.396 

18 

(check)  d* 

68.2 

24.6 

1  06. 

24.9 

*a'  =  tan-i  xly  disregarding  signs  of  x  and  y. 

f  a  =  tan-i  xly,  regarding  the  signs  of  x  and  y,  and  may  be  found  for 
a  given  value  of  a  by  means  of  the  table  of  Art.  44. 


RECTANGULAR    COORDINATES 


17 


25.  From  General  Rectangular  to  Polar  Coordinates. —  We  are 

given  the  general  coordinates  xk ,  yk  of  the  point  Pk ,  and  xn ,  yn  of 
the  point  PH .    We  take  Pk  as  a  pole,  and  -f-  F(axis)  as  zero  direc- 
tion.   Required  the  polar  coordinates,  azimuth,  ak  „  (or  bearing,  &.„) 
and  the  distance  dk.n  of  PH      ^      ~, 
referred  to  Pk ,  Fig.  16.  ^ 

Or,  briefly, 
Given     Pk  (xk,  yk*}, 

Required     Pn  (ak.n  ,   dk  n  ) 
or         PH  (&„,   dk.n}. 

From  Fig.  2,  tan  a,,.n  = 
tan  ^  „  =  (xn  — xk)  lyn — yk 
=  &**.„  /  ^n=x*.JyA.n,  FlG'16 

where  xA.n ,  yk.H  are  the  local  rectangular  coordinates  of  Pn ,  Pk 
being  the  origin.  This  problem  is,  therefore,  after  the  x-  and  y- 
differences  are  found,  like  that  of  Art.  24. 


18  COORDINATES    OF    ELEMENTARY    SURVEYING 


CHAPTER  V. 
POLAR  COORDINATES. 

26.  Polar  Coordinates.— In  Fig.  17  is  shown  a  point  Preferred 
to  the  polar  axis  or  initial  line  OR  and  the  pole  O  by  the  polar 

r\  coordinates  6  and  p.     6  is  the  vectorial  angle 

/'        N\  and  p  is  the  radius  vector.    0  increases  in  counter- 

/   Q  ^_ L_^.F\  clockwise  direction.     Such  are  the  notation  and 

\        ^^X^  nomenclature  of  coordinate  geometry.     In  this 

p  book  we  shall,   while  using  the  principles  pre- 

pIG  sented  in  coordinate  geometry,  employ  symbols 

and  names  suggested  by  a  consideration  of  the 

commonly  accepted  terms  of  surveying. 

27.  Direction  and  Distance. —  In  Fig.  18,  P9PS  is    the   "zero 
direction ' '  (polar  axis). 

P0  is  the  pole. 

P2  is  any  point. 

00.2  is  the  direction  of  line  P0P.2  (vectorial  angle). 

dQ.s  is  the  distance  P0P2  (radius  vector). 

00..2 ,  */0.2  are  the  polar  coordinates  of  the  point  Pr 

Direction  may  be  considered  to  increase  in  clock- 
wise direction  from  o°  to  360°,  in  which  case  it  is 
called  azimuth,  a  (Art.  28)  ;    to  increase  in  both 
clockwise  and  counter-clockwise  directions  from  o° 
to  90°  (two  opposite  zero  directions  being  employed),  in  which  case 
it  is  called  bearing,  (3  (Art.  29)  ;  or  to  increase  in  both  clockwise 
and  counter-clockwise  directions  from  o°  to  180°,  in  which  case  it  is 
called  deflection,  8  (Art.  30). 

28.  Azimuth  and  Distance. —  When  all  the  points  of  a  survey 
are  determined  by  polar  coordinates  of  which  the  directions  (vectorial 
angles)  all   refer    to   one  zero  direction,    the  directions  are  called 
azimuths.     In  this  book  azimuth  will  be  represented  by  a  and  con- 
sidered to  increase  in  clockwise  direction  from  o°  to  360°  or  more. 
Referring  to  Fig..  19, 

P!  P8  is  the  zero  direction  (or  polar  axis) ; 
Pl  is  the  pole  ; 


POLAR    COORDINATES 


19 


\  is  a  point  in  quadrant  I;  P3  in  quadrant  II;  Pt  in  III;  and  P&  in  IV. 
a. 


is  the  azimuth  and 


d 


is  the  distance  (from 
the  pole)  of 


P.. 


means  "Given  the  azimuth 


The  expression  "  Given  P.z  (arz, 
arv  and  distance  dl  2  of  the 
point  P2  referred  to  the  pole 

PI- 

The  azimuth  of  a  line  is 

the  azimuth  of  the  terminal 

point  of  the  line,  the  initial 

point  of  the  line  serving  as 

the  pole. 

The  zero  direction  for 

azimuth  can  be  the  direction 

of  any  given  directed  line. 

True  azimuth,  /a,  is  azimuth 

for  which  the  zero  direction 

is  either  true  north  or  true 

south.     (In  this  book  true 

north  is  taken  as  the  zero 

azimuth.)     Magnetic  azimuth,  ma,  is  azimuth  for  which  the  zero 

direction    is    magnetic 
north. 

29.  Bearing  and  Dis- 
tance.—In  Fig.  20,  P! 
is  the  pole.  PI  P8  is  one 
zero  direction  which 
serves  for  all  points  in 
quadrants  I  and  IV. 
P,  P9  is  the  other  zero 
direction  and  serves  for 
all  points  in  quadrants 
II  and  III. 

The  directions  for  all 
points  in  quadrants  I 
and  III  are  reckoned 
clockwise,  and  direc- 
tions for  all  points  in  II 
and  IV  are  reckoned 


20 


COORDINATES    OF     ELEMENTARY     SURVEYING 


counter-clockwise.  Therefore  in  order  to  completely  determine  a 
point's  position,  the  data  must  include  its  quadrant.  It  is  the  cus- 
tom to  call  one  zero  direction  "North,"  the  other  "South";  and 
to  indicate  quadrants  I,  II,  III,  IV,  by  NE,  SE,  SW,  NW,  respec- 
tively. The  notation  for  polar  coordinates  when  bearings  are  used 
is  indicated  in  the  following  : 


The  bearing  of 


P, 
P, 


IS 


N/8,,E 
S/8...E 

SA.,W 
N/8,.SW 


the  distance  of 


IS 


The  expression  "Given  P2  (N/3,..,  E,  dr$'  means  " Given  the  bear- 
ing (N$r.2E)  and  distance  G^.,)  of  the  point  7^,  referred  to  the 
pole  TV' 

The  bearing  of  a  line  is  the  bearing  of  the  terminal  point  of  the 
line,  the  initial  point  of  the  line  serving  as  the  pole. 

The  two  zero  directions  for  bearing  can  be  the  two  directions  of 
any  doubly  directed  line.  True  bearing,  //3 ,  has  for  its  zero  direction 
either  true  north  or  true  south.  Magnetic  bearing,  m/3 ,  has  for  its 
zero  direction  either  magnetic  north  or  magnetic  south.  The  true 
bearing  of  magnetic  north  is  the  declination  of  the  magnetic  needle. 

30.  Deflection  and  Distance —  Given  any  three  points  Pfc ,  PH , 
TV,  then  the  angle  (not  greater  than  180°)  between  the  line  PnPr 
and  the  line  PkPn  produced  is  called  a  deflection  angle,  or  deflection, 
and  in  this  book  will  be  represented  by  8. 


If  TV  lies  on  the 


right 
left 


written 


right 
left 


the 


and 


deflection  is  called  deflection 
SR; 
SL. 

In  Fig.  21,  of  the  polar  coordinates  of 
7*2 ,  Pl  is  the  pole  ;  P9  Pl  is  the  zero  direc- 
tion (polar  axis);  and  the  coordinates  are 
£j.2  R,  the  deflection  right  (vectorial  angle) ; 
and  </1>2,  the  distance  (radius  vector). 

The  expression  "Given  PM(B^aRt  */*.„)"  means 
angle  ^>^n  between  Pk  Pn  and  the  preceding  line  produced  (the  angle 
§fc.n  being  on  the  right  side  of  the  latter),  and  the  horizontal  dis- 
tance dL  „." 


FIG.  21 


Given  the 


POLAR    COORDINATES 


21 


31.  Angle  Between  Two  Lines:   Azimuths.  —  Given   the  azi- 
muths (referred  to  one  zero  direction)  of  two  lines, 


Pr  P, 

the   angle    between  the  two  lines  (following  in  clockwise  direction 
the  first  line),  is  /j 

?=    <*T.S    —    0,&n> 

If  the  azimuth  of  the  first  line  is  less  than  that  of  the  second  line 
add  360°  to  the  azimuth  of  the  first  line  before  subtracting. 

32.  Angle  Between  Two  Lines:  Bearings.  —  The  following 
table  is  constructed  on  the  supposition  that  the  angle  sought  follows 
(in  clockwise  direction)  the  line  whose  quadrant  is  given  at  the  top 
of  the  table. 

ANGLE  BETWEEN  Two  LINES  OF  KNOWN  BEARINGS. 


NE 

SE 

s  w 

N  W 

NE 

diff. 

1  80°  •-(-  sum 

i8o°  +  NE-SW 

sum  * 

SE 

1  80°  —  sum 

diff. 

360°  -—  sum 

i8o°  +  NW-SE 

SW 

i8o°-fSW-NE 

sum 

diff. 

1  80°  -f-  sum 

NW 

360°  —  sum 

i8o°+SE-NW 

1  80°  —  sum 

diff. 

Example.  — What  is  the  angle  (measured  in  clockwise  direction 


from  first  named  line)  between  a  line  of 


NVV 
SE 


SE 

NW 


bearing  and  a  line  of 


bearing  ? 


We  enter  column 


SE 
NW 


and  follow  down  to  line 


NW, 
SE, 


where 


we  find 


1 80°  +  SE  — NW, 
1 80°  4-  NE  — SW, 


which   means   that  for  the  sought 


angle  we  are  to  add  180°  to  the 
NW 


subtract  the 


SW 


SE 
NE 


bearing  and  from  the  sum 


bearing. 


*  "  Sum"  means  "sum  of  given  bearings,"  and  "  diff."  means  "  differ- 
ence of  given  bearings." 


22 


COORDINATES    OF     ELEMENTARY     SURVEYING 


33.  Distance  Between  Two  Points. —  We  are  given  the  azimuth 
ak  and  the  distance  dk  of  the  point  Pk ;    and  the  azimuth  an  and 

the  distance  dn  of  the  point  Pn .     Both 
points   are   referred   to   the  same    pole. 
Required   the  distance  dk.H  between  Pk 
and  Pn .     Or,  briefly, 
Given      Pk(ak,  ^),     PM(aH,    dn\ 
Required  dk.H. 

The  required  distance  is 

dk.n  =  \d£  -f  d*  —  2d*  dn  cos  Aa,.,,]* 
where     Aax..w  =  azimuth  difference 

==  «„  —  «>t  • 

If  an  <  a, ,  add  360°  to  an  before  sub- 
tracting ak . 

If  Aa^.M  >  90°  and  <  270°,  its  cosine 
is  negative. 

The  following  table  may  be  of  use 
where  the  computation  for  d^.n  is  made 
with  a  slide-rule,  or  with  a  table  of  logarithmic  cosines  in  which  the 
angles  are  written  for  the  first  quadrant  only. 


EQUIVALENTS  OF  Cos  Aa. 


When  Aa  lies  in 

Quad.    I 

Quad.  II 

Quad.  Ill 

Quad.  IV 

cos  Aa 

*sin  (90°  —  Aa) 

—  sin  (Aa-9o°) 

—cos  (Aa-i8o°) 
*-sin  (27O°-Aa) 

sin  (Aa-270°) 

Example.  —  Two  points,  P.z  and  P.A ,  are  referred  to  the  same  pole. 
Given  P.z  (40°,  20),  P3  (350°,  40),  the  directions  being  azimuths. 
What  is  the  distance  between  P2  and  Pz  ? 


*  The  starred  values  of  cos  Aa  are  given  for  slide-rule  use,  since  that 
instrument  is  not  figured  for  cosines. 


POLAR     COORDINATES 


23 


I 

dk 

20. 

2 

dn 

40. 

3 

ak 

40° 

4 

an 

350o° 

5 

Aa  =  an  —  ak 

6 

log  cos  Aa 

9.808 

7 

log^ 

1.301 

8 

log<4 

i  .602 

9 

log  2 

0.301 

10 

sum  log 

3.012 

ii 

log  d? 

2.602 

12 

log  d* 

3-204 

13 

dk 

400. 

14 

dn 

1600. 

15 

df  +  d* 

2000. 

16 

2dk  dn  cos  Aa 

1030. 

17 

dk-n 

970. 

18 

log  d»; 

2.988 

19 

log  dk.n 

1.494 

20 

d*.n 

31.2 

A 

B 

34.  Area  of  Polygon.—  Given  the  azimuths*  and  distances  (re- 
ferred to  one  pole)  of  a  series  of  points  Plt    P2,    P3  .  .  .  Pn)  taken 
in  order.    Required  the  area  Q  of  the  polygon  bounded  by  the  lines 
/>!/>,    P,PS9    P3P<,   .  .  .  Pn-iPn,    PnP^     Or,  using  symbols, 
Given  Pl  (aiy    */,), 


Required  the  area  Q  of  the  polygon  P1P.,P^  .  .  .  Pn  Pl . 
In  Coordinate  Geometry  it  is  shown  that  the  area  is  : 

Q  =  Y-2.  (d^d.,  sin  Aa1>2  +  d^  sin  Aa,.s  -+•  .  .  . 

4-  dn_i  d,ts\n  Aaw_rw  +  dnd^  sin  AaM>1) 
where  Aa,.2  =  a.2  —  al ,    Aa2.3  =  a3  —  a^ ,    etc. 

Add  360°  to  an  azimuth  when  it  is  less  than  the  preceding  azi- 
muth, before  making  the  subtraction  for  azimuth-difference.  E.  g. , 

*  If  bearings  or  deflections  instead  of  azimuths  are  given,  compute  the 
corresponding  azimuths  (Arts.  39  and  41)  before  beginning  the  work  of  find- 
ing the  area. 


24 


COORDINATES    OF     ELEMENTARY    SURVEYING 


if  a2  <a1,  add  360°  to  a2  before  subtracting  a^.  The  sine  of  an 
angle  between  180°  and  360°  is  negative.  The  preceding  applies  to 
the  equation  above  and  to  the  corresponding  rule,  which  is  : 

To  find  the  area  :    i.  Multiply  the  distance  of  each  point  by  sin 
(azimuth  of  the  following*  point  minus  azimuth  of  the  point  itself), 
and  multiply  this  product  by  the  distance  of  the  following*  point. 
2.  Take  one-half  the  algebraic  sum  of  the  products  for  the  area. 
Example.  —  The  five  corners  of  a  field  are,  taken  in  order, 
^(10°,    12),      P.2(6o°,   8), 
P3(i4o°,    15),    P4(230°,    16), 

P5(34o°,    24), 

the  polar  coordinates  being  azimuths  and  distances  referred  to  one 
pole.  What  is  the  area  of  the  field  ? 


I 

log  dm 

dm-i  dm 

Sin 

2 

Point 

Az. 
a 

Dist. 
d 

Az.-diff. 
Aa 

log  sin  Aa 

Sum 
logs. 

:(+) 

AaOT_,.OT 

(—  ) 

log  dm-T, 

3 

f\ 

10° 

12 

1.079 

4 

130° 

9.884 

2.139 

138 

5 

P. 

140 

15 

i  .  176 

6 

28of 

9-993nt 

2.072n 

118 

7 

Pz 

60 

8 

0.903 

8 

170 

9  .  240 

1-347 

22 

9 

P< 

230 

16 

i  .204 

10 

no 

9-973 

2-557 

36l 

ii 

P. 

340 

24 

1.380 

12 

30 

9.699 

2.158 

144 

J3 

P, 

IO 

12 

1.079 

14 

sums 

665 

118 

15 

118 

16 

2  (area) 

547 

17 

area 

273 

A 

B 

c 

D 

E 

F 

G 

H 

35.  To  Change  Zero  Direction:  Azimuths. —  Given  the  azimuth 
an  of  the  point  Pn ,  and  the  azimuth  at  of  the  point  // .  Required 
the  new  azimuth  aln  of  Pn  when  the  direction  pole-P/  is  taken  as 
the  new  zero  direction. 


*  The  point  following  Pn  is  Pl . 

t  a3  —  a.2  =  60°  -f-  360°  —  140°  =  280°. 


Jsin  280°  =  — sin  80° 


POLAR    COORDINATES 


25 


=  a,t — at 


in  which  an  is  to  be  increased  by  360°  if 


The 


north  =  360 


true 

magnetic 
_  I  mag. 


true 


azimuth  of 


azimuth  of 


mag. 
true 
true 
mag. 


north  =  declination  of  the  needle. 

Example. —  The  magnetic  azimuth  of 
true  north  is  345°.  The  magnetic  azimuth 
of  a  line  is  287°.  What  is  the  true  azi- 
muth (Art.  28)  of  the  line  ? 

True  az.  =  mag.  az.  of  given  line  minus 
mag.  az.  of  new  zero  direction  =  287°  (+  360°)  --  345°  =  302°. 

36.  To  Change  Zero  Direction :  Bearings. —  Given  the  magnetic 
bearing  m/3  of  a  line  and  the  declination  A  of  the  magnetic  needle. 
(The  declination  of  the  magnetic  needle  is  the  true  bearing  of  the 
magnetic  north.)  The  true  bearing  t$  corresponding  to  m(3  is 
found  from  the  following  two  tables  : 

Table  A. —  FROM  MAGNETIC  TO  TRUE  BEARING, 

For  use  where  True  Bearing  of  Magnetic  North  is  NA°  E, 

i.  e. ,  where  Declination  of  Needle  is  A°  East. 


NE 


SE 


sw 


NW 


NE 


SE 


Iffl7j?+A>90° 
(M80-M+4) 


SW 


IfmJKA 


NW 


26 


COORDINATES     OF     ELEMENTARY     SURVEYING 


Table  B. —  FROM  MAGNETIC  TO  TRUE  BEARING, 

For  use  where  True  Bearing  of  Magnetic  North  is  NA°  W, 

/'.  e. ,  where  Declination  of  Needle  is  A°  West. 


NE 


SE 


sw 


NW 


NE 


SE 


SW 


NW 


To  use  either  table  follow  down  the  column  headed  with  the  quad- 
rant of  the  given  magnetic  bearing  m/3  till  the  "If"  .  .  .  statement 
is  satisfied  by  the  given  bearing,  and  use  the  equation  for  t$  there 
found  and  that  quadrant  which  stands  on  the  same  horizontal  line. 

It  will  be  evident  that  the  tables  serve  for  changing  from  true  to 
magnetic  bearing  by  interchanging  the  words  and  symbols  for 
"true"  and  "magnetic"  in  the  tables  and  interchanging  the  tables. 

Example. — The  magnetic  bearing  of  a  line  is  S  82°  E  and  the 
declination  of  the  needle  is  25°  East.  What  is  the  true  bearing  of 
the  line? 

The  declination  of  the  needle  being  East,  we  use  Table  A.  We  run 
down  column  '  *  SE ' '  and  find  that  the  given  bearing  and  declination 
satisfy  the  condition  "If  mfl  >  A"  (/.  e.,  82°  >  25°).  We  there- 
fore use  the  equation  there  found  :  t$  —  m@  —  A  (z.  e.>  t$  =  82° 
—  25°  =  57°),  and  quadrant  SE  which  is  named  on  the  same  hori- 
zontal line.  Thus  we  find  the  required  bearing  is  S  57°  E. 

Example  2. — The  true  bearing  of  a  line  is  N  80°  W,  and  the  de- 
clination of  the  needle  is  15°  W.  What  is  the  magnetic  bearing  of 
the  line  ? 

If  the  problem  were  to  change  from  magnetic  to  true  bearing  we 
should  use  Table  B,  but  for  the  problem  as  stated  we  interchange 


I    UNIVERSITY 

V  OF  / 

X^^UFosSifc^ 


POLAR    COORDINATES  27 

the  tables,  2.  e.  ,  use  Table  A  ;  and  in  this  table  read  //3  for  m/3  and 
m(B  for  //3.  The  true  bearing  of  the  given  line  being  N  80°  W  we 
enter  Table  A  and  find  in  ~—-*4*-  ___ 

column   "NW"    that  the  ^^     ^To^^^n 

data   satisfy  the   condition  &/*"  °O 

"If  m/3  >  A"   (it  being       <jf  V 

IfA».     We 


accordingly  use  the  equa-    f  M  \ 

tion  m/3  =  i$  —  A  (which   /  |  * 

is  the  equation  there  found,  -J-2.~IO°  --  W~  +  —  EL  --  90°  4- 
with    w/3    and    t$    inter-   \  I  I 

changed)  and  the  quadrant     \  |  / 

NW  found   on    the  same      \  P=  OC-I6O°     p=l60'-OC       I 
horizontal  line.      We  thus        Q  £y 

find  the  required  magnetic  ^ 

bearing  to  be  m/3  =  N  (80°  '#  ^ 

—  150)  W  =  N  65°  W. 

37.  From  Azimuth  to  FIG.  24 

Bearing.  —  The  diagram,  Fig.  24,  gives  the  bearing  /3  in  terms  of 
azimuth  a  for  each  quadrant.  The  diagram  assumes  that  the  azi- 
muth of  N  is  zero. 

Example.  —  The  true  azimuths  of  certain  lines  are:   35°,  127°, 
230°,  290°.     What  are  the  corresponding  bearings  ?     By  use  of  the 

diagram  we  make  out   this 


a  (Given) 


35 
127° 
230° 

290C 


(Required) 


S  50  W 

N  70  W 


table. 


38.  From  Azimuth  to 
Deflection. —  Given  the  azi- 
muth a0.k  for  the  point  Pfc , 
and  the  azimuth  a0  n  for  the 
point  Pn .  The  zero  direc- 
tion for  deflection  (Art.  30) 
is  P0  Pk .  Required  B0.H ,  the  deflection  of  point  Pu . 

£10.;2  =  a0.H  —  do'ii" 

If  a0.n  <  a0.k ,  add  360°  to  a0.n  before  making  the  subtraction. 
If«l,.*<  1 80°,    5VH  =  8,.Mtf. 
If  &0.M  >  1 80°,    360°  —  B10.M  =  B0.H  L. 

Example. —  The  azimuth  of  P^P2  is  210°,  of  P1  P3  is  10°.  If 
P1P2  is  taken  as  the  zero  direction  for  deflection,  what  is  the  deflec- 
tion of  PJ 


28  COORDINATES    OF    ELEMENTARY    SURVEYING 


-^  0\.3  =   10°  -f   360° 210° 

360°      ^Qy  —  I6o°  <  I8o° 


f>    I         CX^P           \  39»   From  Bearing   to 

N                              \  Azimuth.  — The  diagram, 

T  terms  of  bearing,  for  each 

\                               5                               '  quadrant.    The  diagram  as- 

\    (X  =  \80e4G     '    a  =  l80q-f3     /  sumes  that  the  bearing  of 

Q                                                    ^  zero  azimuth  is  N. 

^o                                             Jy  Example.  —  The   mag- 

ty  o  O 

J&              160             O^  netic  bearings  (Art   29)  of 

"-- —  ,,L  - — •""  certain  lines  are  N  35°  E, 

FlG.25  S53°E,  S5o°W,  N7o°W. 
What  are  the  correspond- 
ing magnetic  azimuths?     By  use  of  Fig.  25  we  make  out  the  table  : 

40.  From  Bearings  to 
Deflection.  —  Given  /3Z  = 


fi  (Given) 


N  35°  E 
S53°E 
S  50°  W 
N  70°  W 


a  (Required) 


35 
127° 
230° 


bearing  of  zero  direction  for 
deflection  ;  /3n  =  bearing  of 
any  other  line.  Required  £„ 
=  corresponding  deflection 
of  other  line.  R  indicates 
deflection  right,  and  L  left. 

The  required  deflection  is  found  by  means  of  Table  C,  page  29. 

To  use  the  table  :  In  the  column  headed  with  the  quadrant  of  the 
given  bearing  of  the  zero  deflection  line,  and  on  the  line  begun  with 
the  quadrant  of  the  given  bearing  of  the  other  line,  note  which  con- 
dition there  written  is  satisfied  by  the  given  bearings,  and  use  the 
equation  for  8  written  immediately  below. 

Example.  —  The  bearing  of  a  line  a  is  N  40°  W,  and  that  of  a  line 
b  is  S  50°  E.  What  is  the  deflection  of  the  line  b  referred  to  the 
line  a  ? 

In  Table  C,  in  column  "NW"  and  on  line  "  SE  "  we  find  two 
conditions  :  ' '  If  ft  >  &/  'and  "  If  &<&«•"  The  data  satisfy  the 
second  condition  (i.  e.,  40°  <  50°),  and  we  therefore  use  the  second 
equation  :  S  =  (180°  -f-  fiz  —  &,)  R  =  (180°  -f  40°  —  50°)  R. 
The  required  deflection  is,  therefore,  B  =  170°  R. 


POLAR    COORDINATES 


29 


Table  C. —  BEARING  TO  DEFLECTION. 


QUADRANT  or  j8z 


NE 


SE 


sw 


NW 


NE 


5-&-JUL 


5=180-1 


4 


SE 


H80+|5rjyR 


sw 


S=(l80+j3n-j3z)R 


NW 


lfJ5z>j3n 

&=(l80°+j3n-jyL 


41.  From  Deflection  to  Azimuth. —  The  deflection  8n  of  a  line  is 
given,  and  the  azimuth  az  of  the  zero  direction  for  deflection  is  also 
given.  Required  the  azimuth  an  of  the  line. 

an  =  az  ^_  g"  L. 
Or  in  words  :     If  the  deflection  is 

muth  of  the  zero  deflection  line,  the  given  deflection  to  obtain  the 
required  azimuth. 

If  az  <  8n  L  add  360°  to  az  before  making  the  subtraction.  If 
(az  -f-  &UR)  >  360°,  decrease  the  sum  by  360°. 


R 
L 
rig! 
left, 

in 
it, 

dicates  deflectior 

add  to 
subtract  from 

i 

1 

ri^ht. 
left. 

;he  azi- 

30 


COORDINATES    OF     ELEMENTARY    SURVEYING 


Example.  —  The  deflection  of  line  b,  referred  to  line  a,  is  153°  L. 
The  azimuth  of  line  a  is  327°.     What  is  the  azimuth  of  line  bl 
Az.  of  b  =  327°    -  153°  =  174°. 

42.  From  Deflection  to  Bearing.— The  deflection  of  a  line  is  5, 
and  the  bearing  of  the  zero  deflection  line  is  & .  Required  the  bear- 
ing /3  corresponding  to  8. 

For  the  solution  of  this  problem  the  following  two  tables  are 
given,  one  for  use  when  the  given  deflection  is  right,  and  the  other 
when  it  is  left. 

Table  D.  —  DEFLECTION  RIGHT  TO  BEARING. 


1  QUADRANT  OF  fin 

QUADRANT  OF  J3Z 

NE 

SE 

SW 

NW 

NE 

IfS<90-j3z 
frfrrf 

If  8>!80-J3Z 
j3=]32+6-l80 

IfS>J3z 
<90+J3Z 
J3-S-J3Z 

If£=90-J3z 
J3=90 

IfS=l80-J3z 
^=0 

SE 

IfS>90-J3z 
<  I80-J3Z 
JH80-(SZ+S) 

IfS<j32 
/3-j3z-S 

ffS>90+0z 
J3H80+&-6 

If<5=90+)32 
J3-90 

SW 

If  S>  I80-& 
J3=J3Z+H80 

If£>j3z 
<90+j3z 
£-$-& 

If6<90-|3z 
^5 

ffW8(HBz 
J3=0 

If6=90-j32 
J3=90 

NW 

If6>90+J3z 
J3=jSz+l80-«S 

If6>90-J3z 
<I80-J3Z 
J3=I80-(J3Z+^) 

Ifi<j3z 
J3=^z-^ 

If  6=90+J3Z 
j6=90 

To  use  Table  D  or  Table  E  :    In  the  column  headed  with  the 
bearing  of  the  zero  deflection  line,  find  the  condition  which  is  satis- 


POLAR    COORDINATES 


31 


Table  E.  —  DEFLECTION  LEFT  TO  BEARING. 


QUADRANT  OF  J3n  "1 

QUADRANT  OF  j8z 

NE 

SE 

SW 

NW 

NE 

ffS<fe 
JHM 

If6>90-j3z 
<I80-J3Z 
JH80-(fet-6) 

If  S)9(H-J3Z 
JHBO+frS 

If6=90+J32 
j8-90 

SE 

IfS<90-j32 
JW* 

If8>j32 
<90+J3, 
j3=S-^2 

FS>I80-J3Z 
JH+J3ZH80 

If£=90-J32 
J3-90 

IfS-l80-j32 
j8-0 

sw 

IfS>90-4J2 
J3=J3Z-H80-S 

If6<j32 
/3=/32-5 

IfS>90-J3z 
<I80-^Z 
jBH80-(&-6) 

IfS-90+J3z 
J3-90 

NW 

ffi-A 

<90+J3Z 
£=6-J32 

If6>l80-J3z 
£=A+S-I80 

If8<90-j32 

^=^z+8 

If8-I80-J3z 
J3-0 

F6-90-J32 
^-90 

fied  by  the  data,  and  immediately  beneath  this  find  the  required 
equation  for  /3,  and  on  the  same  line  on  the  left  find  the  quadrant 
for/3. 

Example.  —  The  deflection  of  a  line  a  is  160°  R,  and  the  bearing 
of  the  zero  deflection  line  is  S  40°  E.  What  is  the  bearing  of 
line  a  ? 

In  Table  D  in  column  SE  we  find  that  the  condition  :  "If  S  > 
90  -f  &"  is  satisfied  by  the  data  (/.  e.,  160  >  90  +  40).  We 
therefore  use  the  equation  fi  =  $z  -f  180  —  B  there  found,  and  use 
the  quadrant  NW  found  on  the  same  line.  The  bearing  of  line  a  is 
therefore  N  (40  -f  180  —  160)  W,  or  N  60  W. 


3-.  COORDINATES    OF    ELEMENTARY    SURVEYING 

43.  To  Change  Pole  Without  Changing  Zero  Direction:  Azi- 
muths.—  Given  the  azimuth  a0.k  and  distance  d0.k  of  the  point  Pk 
referred  to  the  origin  P0 ;   and  the  azimuth  ak.n  and  distance  dk.n 
of  the  point  Pn  referred  to  the  pole  Pk .     Required  to  find  the  azi- 
muth a0.n  and  distance  d0.n  of  PH  referred  to  the  pole  P0 . 

d0.n  and  dk.n  form  two  sides  of  the  triangle  P0PkPn,  and  the 
angle  60.A.n  included  between  them  is  known  in  terms  of  a0.k  and 
ak.n .  d0.n  is  the  third  side  of  the  triangle,  and  the  problem  re- 
duces to  finding  the  distance  between  two  points,  for  which  see 
Art.  33. 

44.  From  Local  Polar  to  Local  Rectangular  Coordinates :  Azi- 
muths.—  Given  the  azimuth  ak.n  and  the  distance  dk.n  of  the  point  Pn 
referred  to  the  local  pole  Pk .     Take  Pk  for  local  origin  for  rectangu- 
lar coordinates,  making  the  axis  +  Yk  identical  with  the  zero  direc- 
tion for  azimuth.      Required  the  rectangular  coordinates  xk.n ,   yk.n 

of  Pn  referred  to  Pk .     Or,  briefly, 


Given 
Required 


Pn   («*„  , 

/U***i 

=  dk. 
=  d, 


sn  ak.n 
cos  afc.n 


FIG.  26 


the  point  PH  lies, 
this  table. 


The  signs  of  sin  ak  n  and  cos  ak  n 
depend  on  the  quadrant  in  which 
The  resulting  signs  of  xfc.H  and  yk.n  are  given  in 


Quad.  I 

Quad.  II 

Quad.  Ill 

Quad.  IV 

Sign  of  x 
Sign  of  y 

.+ 
+ 

+ 

— 

+ 

Logarithmic  sines  and  cosines  are  preferably  taken  from  a  table 
which  is  figured  at  top  and  bottom  of  each  page  for  the  four  quad- 
rants, as  exemplified  in  Hussey's  Mathematical  Tables.*  When  a 
table  figured  for  only  the  first  quadrant  must  be  used,  functions  of 
first-quadrant  azimuths  which  are  equivalent  to  the  azimuths  of  the 
various  quadrants  are  found  by  means  of  the  following  table. 


*Allynand  Bacon,  Boston. 


POLAR    COORDINATES 


33 


Table  F. —  EQUIVALENT  FIRST-QUADRANT  FUNCTIONS. 


When  oc 
Terminates 
in  Quad. 

i. 

5//7  a= 

Cos  a- 

Tan  OC= 

Cot  a= 

FIRST  QUADRANT  EQU/I/ALENTS 

sin  OL 

cos  a. 

[sin  (90-  a)] 

tan  a 

cot  a 

[Tan  (90-00] 

i. 

cos(cc-9o) 
[sin(iao-CCJ] 

-5in(GC-90) 

-cot(tt-90) 
[-Tan(l80-ttj 

-tan((I-90) 

ft 

-sin  (OH  80) 

-cos(a-90) 
[-sin(£70-d| 

tan(a-ISO) 

cot(OH80) 
[tan(£70-a)] 

K 

-cos(a-c70) 
;-sin(360-al 

sin(d-270) 

-cot(CC-£70) 
[-tan(360-dj 

-tan(tt-c70] 

The.  bracketed  equivalents  are  to  be  used  when  computations  are 
made  with  the  slide-rule,  since  that  instrument  has  only  sine  and 
tangent  scales. 

Example.  —  The  pole  is  Pl .  The  polar  coordinates  of  P.L  are  1 5°, 
16  ;  of  P.A,  200°,  32.  What  are  the  rectangular  coordinates  of  P2 
and  Pz  referred  to  origin  Pl  if  the  azimuth  of  -f-  F  (axis)  is  zero  ? 
(See  solution  In  tabular  form  on  following  page.) 

Note. —  If  log  sin  and  log  cos  are  taken  from  a  table  having  only 
first-quadrant  headings,  insert  a  column  between  cols.  C  and  D,  in 
which  to  write  the  equivalent  first-quadrant  angles  as  found  in  .table. 

45.  From  Local  Pclar  to  Local     _^( 
Rectangular  Coordinates :  Bearings. 
—  Given  the  bearing  /3/i,.M  (with  the 
letters  of  its  quadrant)  and  the  dis- 
tance dk  „  of  the  point  Pn ,  referred  to 
the  local  pole  Pk.     We  take  Pk  for 
local   origin  for  rectangular  coord i-     V 
nates   making   -|-  Y  (axis)   identical  FlG-  27 

with  north.     Required  the  rectangular  coordinates  xk.H ,  yk.t 
referred  to  Pk .     Or,  briefly, 


of  P.. 


34 


COORDINATES    OF    ELEMENTARY    SURVEYING 


Point 

Polar  Coords. 

log.*- 
1  log  sin  a 

I  log  <*       1 

log  cos  a  j 
log  j/ 

Rect.  Coords. 

I 

2 

3 

4 
5 

Az. 
a 

Dist. 
d 

X 

y 

+ 

— 

+ 

— 

6 

7 
8 

9 

10 

P, 

15° 

16 

0.617 

9-4I3 
1  .204 
9.984 
1.188 

4-1 

15-4 

ii 

12 
13 

14 
15 

P, 

200° 

32 

i  .O39n 
9-534^ 
1-505 
9-973n 
i  -478n 

10.9 

30.1 

A 

B 

C 

D 

E 

F 

G 

H 

Given 
Required 


sn 
COS 


The  signs  of  sin  /9fe.w  and  cos  fik.n  depend  on  the  quadrant  in 
which  the  point  Pn  lies,  i.  e,,  on  the  quadrant  letters  of  the  bearing. 
The  resulting  signs  of  xk.n  and  yk.n  are  given  in  the  following  table. 


• 

NE 

SE 

sw 

NW 

Sign  of  x 
Sign  of  y 

+ 

-f 

— 

4- 

For  slide-rule  computation  sin  (90  —  /3)  should  be  substituted 
for  cos  /3  because  the  slide-rule  carries  no  scale  of  cosines. 

Example.  —  The  bearings  and  distances  of  two  points,  referred  to 
Pl  as  a  pole,  are  P,  (N  15°  E,  16),  P3  (S  20°  W,  32).  What  are 
the  rectangular  coordinates  of  P2  and  P3 ,  referred  to  origin  Pl ,  if  the 
bearing  of  -f  Y  (axis)  is  north  ? 


POLAR    COORDINATES 


35 


Polar  Coords. 

Rect.  Coords. 

I 

log^ 

2 

f  log  sin  /3 

X 

y 

3 

Point 

Bearing 

Dist. 

{  log  d         \ 

4 

ft 

d 

log  cos  /3  j 

5 

logy 

+ 

+ 

6 

0.617 

4.1 

7 

9-4I3 

8 

Pa 

N  15°  E 

16 

i  .204 

9 

9.984 

10 

1.188 

1,5.4 

1  1 

1.039 

10.9 

12 

9-534 

13 

P3 

S20°W 

32 

1-505 

14 

9-973 

15 

1.478 

30.1 

A 

B 

C 

D 

E 

F 

G 

H 

46.  From  Local  Polar  to  Local  Rectangular  and  Other  Coordi- 
nates :   Deflections. —  Either  change  from  deflections  to  azimuths 
(Art.  41  ),  and  proceed  as  in  Art.  44 ;  or,  change  from  deflections  to 
bearings  (Art.  42),  and  proceed  as  in  Art.  45. 

47.  From  Polar  Coordinates  to  Biangular  and  Biradial  Co- 
ordinates.—  See  Arts.  51,  52  under  Biangular,  and  Art.  59  under 
Bipolar  Coordinates. 


CHAPTER  VI. 


BIANGULAR  COORDINATES. 

48.  Biangular  Coordinates —  Fig.  28.     In  the  triangle  Pt  P2  P, , 
we  are  given  the  distance  dr.z ,  and  the  angles  #rl.3,   0rr3.     The 
position  of  .P3  with  respect  to  Pl  and  P.2 

is  determined  by  the  data.  #rl.3,  Orv* 
are  the  biangular  coordinates  of  P3. 
dr2  is  the  base,  and  Pl ,  P2  are  the  poles 
of  this  system  of  coordinates. 

For  the  purpose  of  computation  the 
trigonometric  notation  for  a  triangle  will 
be  substituted  for  the  foregoing,  and  in 
such  a  way  that  the  base  will  always  be 
represented  by  c.  FlG.  28 

49.  Distance  Between  Two  Points. — 

Fig.  29.     Given  the  distance  PjP.2,  the  biangular  coordinates  0rl.3, 
#^.3  of  the  point  P3 ,  and  the  biangular  coordinates  02<1.4 ,  0rri  of 

the  point  P4.  Required 
the  distance  PaP4 .    Or, 
briefly, 
Given 
dr.z  —  length  of  base, 


Required     dyi . 

The  figure  is  a  quad- 
rilateral with  diagonals. 
The  lines  of  the  figure 
form  four  triangles 
which  are  here  num- 
bered I,  II,  III  IV,—  I  and  II  being  those  for  which  the  base  Pl  P2 
is  a  common  side;  III  and  IV  being  those  for  which  P3P4  is  a 
common  side.  We  use  the  ABCabc  triangle  notation,  employing  a 


BIANGULAR    COORDINATES 


37 


Solution  of  example  in  Article  49,  page  38. 


I 

Triangle  I 

Triangle  II 

2 

A 

100° 

110° 

3 

B 

35° 

45° 

4 

A  +  B 

155° 

155° 

5 

C 

45° 

25° 

6 

c 

20. 

20. 

7 

logr 

1.301 

I.30I 

8 

log  sin  C 

9.850 

9.626 

9 

log  c  1  sin  C 

1.675 

10 

log  sin  A 

9-993 

9-973 

ii 

log  sin  B 

9-759 

9.850 

12 

log  a 

1.444 

1.648 

13 

log  £ 

i  .210 

1-525 

14 

a 

27.8 

44-5 

15 

b 

16.2 

33-5 

16 

AI        100° 

^n      110° 

17 

•#ii       45° 

£i       35° 

18 

Am     55° 

^iv     75° 

19 

Triangle  III 

Triangle  IV 

20 

^ 

55° 

75° 

21 

£ 

16.2 

33-5 

22 

c 

44-5 

27.8 

23 

log  2 

0.301 

0.301 

24 

log  $ 

I  .  2IO 

1-525 

25 

log  c 

I  .648 

1.444 

26 

log  cos  ^4 

9-759 

9-4^3 

27 

log  2.bc  cos  ^ 

2.918 

2.683 

28 

log  P 

2.420 

3-050 

29 

log  r2 

3.296 

2.888 

30 

$2 

263. 

1  1  20. 

31 

<:2 

1980. 

773- 

32 

b1  -j-  <:2 

2243. 

1893- 

33 

2&T  COS   /4 

828. 

482. 

34 

tf2 

1415. 

1411  . 

35 

logrt2 

3.151 

3-150 

36 

log  tf 

I-576 

1-575 

37 

a 

37-7 

37-6 

A 

B 

C 

38 


COORDINATES    OF    ELEMENTARY    SURVEYING 


subscript  to  designate  the  triangle  to  which  each  part  belongs,  and  • 
distribute  the  letters  as  shown  in  the  figure. 

Notice  that  Ci  =  cllt  all  =  cmt  ^  =  clv  ,  bl  =  blUt  bu=-'  blv  . 

We  employ  the  following  equations  : 

C  =  1  80  —  (A  -{-  B}  .  .  (i)  I  (a  and  b  are  biradial  coordi- 
a  =  (c  I  sin  £7)  sin  A  .  .  (2)  J>  nates  of  the  point  opposite  the 
b  =  (c  I  sin  C}  sin  B  .  .  (3)  j  base  c.  See  Art.  56.) 

The  three  equations  above  apply  to  triangles  I  and  II  ;  the  follow- 
ing applies  to  triangles  III  and  IV. 

a  ==  (b1  -f  c*  —  2bc  cos  A)%      .         .         .     (4) 
It  should  be  remembered  that  an  a  of  (i),  (2),  (3)  is  a  c  in  (4). 

Example.  —  Given  the  following  data,  referring  to  Fig.  29  : 


=  no°,    BI  ==  35° 


U  =  45 


=  100 


=  20  ; 


find  the  distance  P3P4  .     (Solution  is  given  on  page  37.) 

50.  Area  of  the  Triangle.  —  From  Trigonometry  we  have  :  Area 
=  CL  sin  A  sin  B  /  2  sin  C  :  in  which  c  —  base  ;  A,  D  —  biangular 
coordinates  ;    C  =    180  - 

(A+B). 

Example.  —  The  biangu- 
lar coordinates  of  P5  ,  refer- 
red to  base  P7P2,  are  0.,.7.5 
=  30°,  Or  2.5  =  70°,  4.,= 
12.  What  is  the  area  of  the 
triangle  P7P5P,? 

51.  From  Biangular  to 
Polar  Coordinates  (Azimuth 
and  Distance).  —  Fig.    30. 
Given  the  base  PjP2  =  ^.2, 
and  the  biangular  coordinates 

02-i-3  >  01-2-s  °f  tne  Pomt  ^a 
referred  to  this  base,  and  the 
azimuth  ar.2  of  the  line  Pl  P2  . 
Required  the  azimuth  a,.3 
and  the  distance  drz  of  P3 
referred  to  the  pole  Pl  . 
Or,  briefly  :  Given 

Pz  (0ri-s»   0r«-s)     and     ai-2> 
Required         P3  (ar3  ,   ^.3). 


I 

Base 

P  P 

*1*2 

2 

Point  opp.  base 

P, 

3 

P7P2  =  ^ 

12. 

4 

#2.7.5  :=r  ^4 

30° 

5 

#7.2.5  ^^  .5 

70° 

6 

^  +^ 

100° 

7 

C 

80° 

8 

log^ 

1.079 

9 

log^2 

2.158 

10 

log  sin  A 

9.699 

ii 

log  sin  B 

9-973 

12 

sum  log 

2.830 

13 

log  2 

0.301 

14 

log  sin  (7 

9-993 

15 

sum  log 

0.294 

16 

log  area 

2-536 

17 

area 

344- 

A 

B 

BIANGULAR    COORDINATES 


39 


Using  the  ABCabc  triangle  notation,  we  put  c—  base,  A  =  Ori.3t 
B  =  0rv9 .  We  know  that  C  =  180  —  (A  +  B).  The  required 
distance  is 

b  — 

*-*"J.JJ        IS       W4.J.X       J^f        I         ^AJJ.         \^        j  *     —      1 

and  the  required  azimuth  is 


rz  =  c  sin  B  /  sin  C  ; 


in  which  0  is  positive  when  reck- 
oned in  clockwise,  and  negative 
when  reckoned  in  counter-clock- 
wise direction  from  the  base. 

Example.—  The  line  P^  is 
40  ft.  long,  and  its  azimuth  is  240°. 
A  point  P3  is  referred  to  P:P2  as  a  base,  by  the  biangular  coordi- 
nates 02.r3  =  50°,  01<2.3  =  80°.  02<1.8  is  reckoned  in  counter-clock- 
wise direction  from  the  base.  What  is  the  azimuth  and  length  of  the 
lineP1P3? 


I 

BaseEEE^P, 

O    E> 

2 

Point  =  P82 

~p 

3 

base  c 

40. 

4 
5 

biang.    f  A 
coords.  1  B 

50° 
80° 

6 

A+B 

I300 

7 

C 

50° 

8 

log^r 

1.602 

9 

log  sin  B 

9  993 

10 

sum  log 

1-595 

ii 

log  sin  C 

9.884 

12 

log  b 

i.  711 

13 

b  =  dr3 

51-4 

H 

ari 

240° 

15 

0vrz  =  A 

50°  neg. 

16 

«,, 

190° 

A 

B 

52.  From  Biangular  to  Polar  Coordinates  (Bearing  and  Dis- 
tance).—  Fig.  31.  Given  the  base  P1P2  =  dv%  and  the  biangular 
coordinates  0rl.t,  0r.2-9  of  the  point  P3  referred  to  this  base,  and  the 
bearing  /3r2  of  the  line  PjP^  Required  the  bearing  /3r3  and  distance 
drs  of  P3  referred  to  pole  Pt.  Or,  briefly, 


40 


COORDINATES    OF    ELEMENTARY    SURVEYING 


Given      P3(0rr3,  01>2.3)     and     ft.,, 
Required  P,  (ft.,,    ^.,). 

Using  the  ABCabc  triangle  nota- 
tion, we  put   c  =  base,  A  =  #rl.3, 

B  =  em. 

Now       C=  1  80  —  04  -f  £), 
FIG.  31  and      ^  =  ^.3  =  c  sin  .Z?  /  sin  £7. 

ft.3  is  found  by  entering  Table  D  (or  E),  Art.  42,  with  fiz  =  ft.a 
and  $  =  £,.!.,. 

53.  From  Biangular  to  Bira- 
dial  Coordinates:  Single  Point 
(i.  e.  ,  given  one  side  and  two  an- 
gles of  a  triangle,  to  find  the  two 
other  sides).  —  Fig.  32.  Given 
the  length  drz  of  the  base  P,P2  , 
and  the  biangular  coordinates 
$i-2-3  >  ^2-i-3  °f  a  point  P3  referred 
to  this  base.  Required  the  bi- 

radial  coordinates  d       d.rz  of  P3  referred  to  the  same  base.     Or, 

briefly, 

Given    P3(0r2,,    0,^), 
and  base  d^  , 

Required     P,  (^.,,    d^. 

To   enable  us   to   use    the 
ABCabc  triangle   notation   we 


^  Now      C  =  1 80  —  (A  +  £), 

#  =  </2.3  =  (c  I  sin  f)  sin  ^4, 
FIG.  33  ^  =  dr^  =  (c  I  sin  (7)  sin  B> 

and  the  check-equation  is     cos  C  =  (a*  -\-  bz  —  ^2)  /  2ab. 

For  an  example  see  the  following  Article. 

54.  From  Biangular  to  Biradial  Coordinates :  Series  of  Points 
(i.  e.)  to  solve  a  chain  of  triangles). —  In  Fig.  33  we  have  a  series 
of  points  Pl ,  P2 ,  P3 ,  .  .  .  .  Given  the  biangular  coordinates  and 
base  for  each  point  of  the  series,  required  the  biradial  coordinates, 
as  follows : 


BIANGULAR    COORDINATES 


41 


Point 

Base 

(Given) 
Biangular  Coordinates 

(Required) 
Biradial  Coordinates 

P, 
Pt 
P. 
P. 

P,P,  =  d^ 
P.iP,  =  d.i.z 

p.p<^4< 

P.P.  —  aU 

02-i-3  >    01-2-3 

#3-2-4  >      #2-3-4 
#4-3-5,     #3-4-5 
P-4-6>      #4-5-6 

^1-3  >     ^2'3 

^2-4  y    dy± 

d^y     ^4-5 

di.~%  d^~ 

In  other  words, 

Given  two  angles  of  each  triangle  of  a  chain,  and  one  side  of  one 
of  the  triangles,  required  to  compute  all  other  triangle  sides  of  the 
chain. 

1.  Plot  the  points  of  the  series,  using  the  data.     Fig.  34.     (This 
plot  need  be  nothing  more  than  a  mere  sketch  or  diagram  upon 
which  to  write  the  notation,  but  if  made  with  some  care  serves  as  a 
rough  check  on  the  computations.) 

2.  Mark  the  data*  upon  the  plot  as  shown. 

3.  Number  the  triangles  I,  II,  III,  .  .  .  ,  as  shown. 

4.  Letter  the  sides  and  angles  of  each  triangle  with  the  ABCabc 
notation,  using  red  ink,  according  to  this 

scheme  : 

(a)  Place  the  letters  within  the  triangle 

as  shown, 

(b)  Call  the  given  base  c, 

(c}  Call  the  base  of  each  succeeding  tri- 
angle c, 

(</)  Represent  by  a  that  side  of  each 
triangle  which  becomes  the  base 
of  the  succeeding  triangle, 

(e)  The  remaining  triangle  parts  are 
now  put  in  to  conform  to  the  cus- 
tomary triangle  notation,  i.  e.,  so 
that  in  each  triangle  angle  A  lies 
opposite  side  a,  B  lies  opposite 
^,  and  C  lies  opposite  c. 

5.  Use  the  equations  of  Art.  53.     It  should  be  remembered  that 
by  the  foregoing  scheme  of  lettering  each  computed  side  a  becomes 
the  known  base  c  for  the  succeeding  triangle. 


FIG.  34 


*  These  data  are  assumed  to  be  free  from  error. 


42 


COORDINATES    OF    ELEMENTARY    SURVEYING 


Example.  —  Given  the  data  shown  in  Fig.  35.      Required  the 
biradial  coordinates  of  P9  and  P± ;  i.  e.,  required  the  lengths  of  the 

triangle  sides, 
p  pr — r-i ; r » P*  i.   We  letter  the  triangles  of 

i  •»  IX.      A  *  o  l«.  ^   ^     /  •  4. 

the  chain  in  accordance  with  the 
foregoing  scheme  exhibited  in 

Fig.  34- 

2.  We  use  these  equations 
of  Art.  53: 

C--~-  i8o-(A  +B), 

FIG.  35  a  =  (c  I  sin  Q  sin  A, 

b  =  (c  I  sin  Q  sin  B, 
and  check  equation     cos  C  =  (a*  -\-  b'1  —  t2)  /  2.ab. 


I 

Triangle  I 

Triangle  II 

2 

C 

20. 

22.3* 

3 

A 

75° 

30° 

4 

B 

45° 

100° 

5 

A+B 

120° 

130° 

6 

C 

60° 

50° 

7 

log£ 

1.301 

1.348 

8 

log  sin  C 

9-938 

9.884 

9 

log  (>/sin  O 

1-363 

1.464 

10 

log  sin  ^4 

9-985 

9.699 

ii 

log  sin  .# 

9.850 

9-993 

12 

log^ 

1-348 

i  .  163 

13 

log  b 

1.213 

1-457 

H 

log  2 

0.301 

0.301 

15 

# 

22.3 

14-5 

16 

b 

16.3 

28.6 

17 

logtf2 

2.696 

2.326 

18 

log  ^ 

2.426 

2.914 

19 

log  c1 

2  .6O2 

2  .696 

20 

c? 

497- 

212. 

21 

& 

267. 

820. 

22 

a1  -f  £a 

764. 

1032. 

23 

r'2 

400. 

497- 

24 

^2   +    ^3  _  <* 

364- 

535- 

25 

log(«2  +  ^2  —  ^-) 

2.561 

2.728 

26 

log   2^ 

2.862 

2.921 

27 

log  (a'+fr  —  <*)l*ab 

9.699 

9.807 

28 

log  cos  C 

9.699 

9.808 

*  c  of  triangle  II  is  identical  with  a  of  triangle  I. 


BIANGULAR    COORDINATES  43 

55.  From  Biangular  to  Rectangular  and  Other  Coordinates 

We  first  make  the  tranformation  from  biangular  to  polar  coordinates  ; 
and  second,  convert  these  polar  coordinates  into  the  required  co- 
ordinates as  directed  under  Polar  Coordinates,  Chap.  V. 


CHAPTER   VII. 


BIRADIAL  COORDINATES. 

56.  Biradial  Coordinates.— Fig.  36.  In  the  triangle  P^P,, 
we  are  given  the  distance  dr.2  and  the  two  distances  drz ,  drz .  The 
position  of  P3  with  respect  to  Pt  and  P2  is  determined  by  the  data. 

dv^  dTZ  are  the  biradial  coordinates  ofP3. 
dVi  is  the  base,  and  P1  and  P2  are  the  poles  of 
this  system  of  coordinates. 

57.  Area  of  the  Triangle — Trigonome- 
try gives  us 

Q  =  area  =  [s  (s  —  a)  (s  —  b}  (s  —  c)]^ 
in  which     s  =  }4(a  H-  b  -f-  r). 
The  check  equation  is 

(s  —  a)  -f-  (s  —  b}  -f  ( s  —  c)  =  s. 
Example.  —  The  three  sides  of  a  triangle  are  12,  8,  6.     What  is 
the  area  of  the  triangle  ? 

58.  From  Biradial  to  Bian- 
gular  Coordinates.  —  Fig.  37. 
Given  the  biradial  coordinates 
drz ,  d.2.3  of  the  point  P3  referred 
to  the  base  Pl  P2  whose  length 
dr.2  is  also   given.      Required 
the  biangular  coordinates  #2.j.3, 
^.g.g  of  P3  referred  to  the  same 
base.     Or,  briefly, 
Given        the  base  dr^ 
and          P3  G^.g,    dr3), 
Required     P3  (#2.,.3,    Ovrs). 

We  reletter  the  triangle  to 
conform  to  the  ABCabc  triangle 
notation,  calling  the  base  c  as 
usual.  The  distribution  of  the  remaining  letters  is  shown  in  the  figure. 


I 

a 

12 

2 

b 

8 

3 

c 

6 

4 

2S 

26 

5 

S 

13 

6 

s  —  a 

i 

7 

s  —  b 

5 

8 

s  —  c 

7 

9 

(check)  s 

13 

10 

log  s 

i  .  114 

ii 

log  (s  —  a) 

0.000 

12 

log  (s  —  b} 

0.699 

13 

log  (s  —  c} 

0.845 

log  Q> 

2.658 

J5 

log<2 

1.329 

16 

area  =  Q 

21.3 

A 

B 

BIRADIAL    COORDINATES  45 

The  three  sides  of  the  triangle  being  a,  b,  c, 

i.  Compute  s  ==  y^(a  -\-  d  -{-  c)       .     (i) 
I      2.  Computer  —  a,  s  —  b,  s — c 

and  solve  the  check  equation 
i?  s=(s  —  a)  +  (s  —  b)  +  (s  —  c)      (2) 

>      3.  Compute 

k  =  [>/(j  -a)(s  —  b)  (s  —  eft*     (3) 
4.   Find  the  angles  A,  B,  C,  from 


FIG.  37 
and  solve  the  check  equation 

•cot  y2B  -  cot  y2c 


cot 


which  is  derived  thus  :  multiply 
together  (4)  ,  (5)  and  (6),  ob- 
taining 

cot  y2A  •  cot  y2B  -  cot  y2  c 

=  fr(s  —  a)  (s  —  b  )  0  —  c) 
=  k*  [k(s  —  a)  (s  —  fi)(s  —  cft 

Replace  ft?  by  its  value  in  (3), 
obtaining  (7). 

The  required  biangular  coordi- 
nates are 

Ori.*  =  A,     Om  =  B. 

Example.  —  Given  the  bira- 
dial  coordinates  d^  =  12,  d,^ 
=  20  of  the  point  P3  referred 
to  the  base  P1P.2  whose  length 
is  1  6.  Required  the  biangular 
coordinates  0rl.3,  0rrB  of  P3 
referred  to  the  same  base. 

59.  From  Biradial  to  Polar 
Coordinates.—  We  (i)  convert 
the  biradial  into  biangular  co- 
ordinates (Art.  58);  and  (2)  con- 
vert the  biangular  coordinates 
into  either  azimuth  or  bearing, 


cot%A=k(s  —  d)          .     (4) 
cot  y2B  =k  (s  —  b)          .     (5) 

i 

a 

20. 

2 

b 

12. 

3 

c 

16. 

4 

2S 

48. 

5 

S 

24. 

6 

s  —  a 

4- 

7 

s-b 

12  . 

8 

s  —  c 

8. 

9 

(check)  s 

24. 

10 

log  (s  —  a) 

0.602 

ii 

log  (s  —  b} 

1.079 

12 

log  (s  —  e) 

0.903 

13 

sum  log 

2-584 

H 

log  s 

1.380 

15 

log  k'- 

8.796 

16 

log  k 

9-  398 

17 

log  cot  y2A 

o.ooo 

18 

log  cot  ^£ 

0.477 

19 

log  cot  y2  c 

0.301 

20 

sum  log 

0.778 

21 

(check)  log  ks 

0.778 

22 

y?A 

45° 

23 

%B 

i8°.4 

24 

y-zC 

26°.  6 

25 

A 

90° 

26 

B 

36°.  8 

27 

C 

53°.  2 

28 

check  :A+B-\-C 

i8o°.o 

A 

B 

46  COORDINATES    OF    ELEMENTARY    SURVEYING 

as  desired.  We  then  have  from  the  given  biradial  coordinates  and 
the  derived  azimuths  (or  bearings)  two  sets  of  polar  coordinates  for 
the  point  P3 ,  viz. : 

PS  (ai'3>    ^1-3)  referred  to  the  pole  P1} 
P3  (a2.3 ,    </2.3)  referred  to  the  pole  P2 ; 

in  which  a  is  azimuth,  and  is  to  be  replaced  by  ft  if  bearing  instead 
of  azimuth  has  been  found. 

60.  From  Biradial  to  Rectangular  and  Other  Coordinates. — 
We  first  derive  polar  coordinates  from  the  data.  The  subsequent 
transformation  from  the  derived  polar  coordinates  into  the  coordi- 
nates desired  is  a  problem  under  Chap.  V. 


CHAPTER  VIII. 


BIPOLAR  COORDINATES. 

61.  Bipolar  Coordinates. —  Given  in  triangle  P^P^P3J  Fig.  38, 
the  length  drz  of  the  side  P^P* ;  the  length  d2.z  of  the  side  P2P3 ; 
and  the  angle  $2-i-3  opposite  P2P3.     The  po- 
sition of  P3  with  respect  to  Pl  and  P2  is  deter- 
mined by  the  data.     #rl.3,  <^2.3  are  the  bipolar 

coordinates  of  P3.    dr2  is  the  base,  and  P1PZ 
are  the  poles  of  this  system  of  coordinates. 

62.  From  Bipolar  to  Polar  Coordinates. — 
Fig-  39-     Given  the  bipolar  coordinates  0yl.* 
and  dTZ  referred  to  the  base  PjP2  whose  given 

length  is  ^r2.     Required  the  polar  coordinates  0r2.3  and  d.rz  of  P3 
referred  to  the  zero  direction  P2PX  and  the  pole  P2.     Or,  briefly, 

Given      drz  =  the  length  of  the  base, 

Required       P3  (@rrs ,    «2.3). 

We  make  use  of  the  ABCabc  triangle 
notation,  calling  the  base  c  and  the  given 
angle  A,  as  shown  in  the  figure.  In  the 

triangle, 

sin  C  =  sin  A  cja 

p    and  the  required   polar  coordinates  are 
0ra.8  =  B  =  1 80  —  (A  +  O> 

Example. — PjP2  is  20  ft.  long.  The  angle  6.2.r3  =  40°,  P2P3 
is  25  ft.  What  are  the  polar  coordinates  0r2.3,  <^2.3  of  Ps  referred 
to  the  base  P,  P2  ? 

We  put     PiP2  =  c,     d.rs  =  a,     Orr$  =  A,     and  so  on. 

(Solution  in  tabular  form  given  on  next  page.) 

63.  From  Bipolar  to  Rectangular  and  Other  Coordinates — We 
convert  the  bipolar  into  polar  coordinates  (Art.  62).     The  derived 
polar  coordinates  are  converted  into  rectangular  or  other  coordinates 
as  shown  in  Chap.  V. 


48  COORDINATES    OF    ELEMENTARY    SURVEYING 

Solution  of  Example  in  Article  62,  page  47  : 


I 

A 

40° 

2 

c 

20 

3 

a 

25 

4 

log^ 

I  .301 

5 

log  sin  A 

9.808 

6 

log  c  sin  A 

I  .  lOQ 

7 

log  a 

1.398 

8 

log  sin  C 

9.7II 

9 

C 

30°.  9 

10 

A  +  C 

70°.  9 

ii 

B 

109°.  i 

A 

B 

CHAPTER   IX. 
TRIPOLAR  COORDINATES. 

64.  Tripolar  Coordinates.  —  Fig.  40.  Given  the  distances 
P:P2  =  c,  and  P.2P3  =  c'9  angle  P^P^P*  =  6.  Given,  also,  the 
two  angles  Pl  P,  P2 ,  P2  P4  P3 .  These  data 
are  sufficient  to  determine  the  position  of 
the  point  P4  with  respect  to  Plt  P.2,  P3 
(except  when  P4  falls  on  the  circumference 
drawn  through  P, ,  P2 ,  P3). 

C,  C'  are  the  tripolar  coordinates  of  P4 


referred  to  the 


P  P  —  P  P 


65.  From  Tripolar  to  Polar  Coordi- 
nates. —  Fig.  41.  Given  the  two  segments 
PlP2  =  c,  P.,P3  =  c',  and  the  included 
angle  6,  of  the  double  base  PlPt  —  P2P3. 


FIG.  41 


Given  also  the  tripolar 
coordinates  C,  C'  of  the  point  P4 
referred  to  this  double  base.  Re- 
quired the  polar  coordinates  (de- 
flection S,.4  =  B  and  distance  dr4: 
=  a),  of  the  point  P4  referred  to 
zero  direction  P2  Pj  and  pole  P2 . 

We  use  the  ABCabc  triangle 
notation  as  shown  in  Fig.  41  where 
the  letters  of  the  right-hand  tri- 
angle are  primed,  a  and  a'  are 
identical. 
a  =  a '  =  c  sin  A  /  sin  C 

=  c'  sin  A'  /sin  C'     (i) 


A  +  A'  =  360  -  (C  +  C  +  0)  - 
From  (2)  we  have 


and       sin  A'  =  sin  (7  —  A}  =  sin  7  cos  A  —  cos  7  sin  A  .     (2') 


50 


COORDINATES    OF    ELEMENTARY    SURVEYING 


Substituting  this  value  of  sin  A'  in  (i),  we  obtain 

c  sin  A  I  sin  C  --  (cr  sin  7  cos  A  —  c'  cos  7  sin  A)  /  sin  C' 
or,  cot  A  =  (c  sin  C'  /sin  C  +  ^'  cos  7)  /  r'  sin  7     .     (3) 


where 

M  —  csm  £"/sin 
N  =  c'  cos  7 
O  =  c'  sin  7 


(2') 
(4) 
(4) 


I 

c 

20 

2 

c' 

40 

3 

C 

15° 

4 

C' 

28° 

5 

6 

140° 

6 

c  'H-  c*  -\-  o 

183° 

7 

7 

177° 

8 

log  c 

1.301 

9 

log  sin  C' 

9.672 

10 

sum  log 

0-973 

ii 

log  sin  C 

9-4I3 

12 

log  M 

i  .560 

13 

log  N 

i  .6oin 

14 

log  cos  7 

9-999* 

15 

log  c[ 

i  .602 

16 

log  sin  7 

8.719 

17 

log  (9 

0.321 

18 

Jf 

36.3 

19 

N 

—39  9 

20 

M  +  N 

—  3-6 

21 

log  M+  N 

o  .  E\56n 

22 

log  cot  ^4 

0.23  5* 

23 

^4 

149°.  8   * 

24 

7  —  A  =  A  ' 

27°.  2 

25 

A       \       /~* 
JT±    ~p     tx 

164°.  8 

26 

B 

15°.  2 

27 

A'  +  C' 

55°  -2 

28 

B' 

124°.  8 

29 

log  sin  A 

9.702 

3° 

log  £  sin  ^4 

1.003 

31 

log  ,4 

1-590 

S2 

log  sin  A  ' 

9.660 

33 

log  c'  sin  ^4  ' 

i  .262 

34 

check  :  log  a  ' 

1-59° 

35 

a' 

38.9 

#==  I8o  —  (A  +  C) 
#  =  =180—04'+  C") 

The  polar  coordinates  (de- 
flection and  distance)  are,  there- 
fore, respectively, 

*n  =  B  .  .  (5) 
d>2-i  =  a  =  c  sin  A  /sin  C  (6) 
the  zero  direction  being  P.iPl 
and  the  pole  P2  . 

The  check  equation  is 
dri  =  a'  =  c'smAf  /sin  C'  (6') 

There  are  two  possible  po- 
sitions of  the  point  P4,  one  on 
the  concave  side  of  the  double 
base,  and  the  other  on  the  con- 
vex side. 

Example.  —  The  point  P4  is 
referred  by  tripolar  coordinates 
to  points  Pt  P2  P3  .  Given  C  = 

15°,  r  =  28-,  p,p>p3  =  e 

=  I40°,    P1P,=  *=20,    P,P. 

=  £•'  =  40.  What  is  the  angle 
pipipi==jg=:5r4i  and  the 
distance  ^2.4  =  a  ? 

Plot  the  points  Px  ,  Pa  ,  P3  on 
one  piece  of  paper.  Draw  on  a 
piece  of  tracing-paper  three  lines  from  any  point  P  so  as  to  form  two 
adjacent  angles  equal  respectively  to  the  given  angles  C  and  C  '. 
Move  the  tracing-paper  about  on  the  plot  of  Plt  Pa  ,  P3  until  the  lines 


TRIPOLAR    COORDINATES  51 

radiating  from  P  respectively  pass  through  Pv  P2,  P3  simulta- 
neously. Prick  through  the  tracing-paper  at  P  on  to  the  other 
paper.  The  pricked  point  is  Pv  Measure  B  and  a  to  check  the 
computations. 

66.  From  Tripolar  to  Rectangular  and  Other  Coordinates. — 
The  tripolar  coordinates  are  first  converted  into  polar  coordinates 
after  which  any  other  desired  coordinates  are  derived  according  to 
the  directions  given  under  Polar  Coordinates,  Chap.  V. 


PART   III.— VERTICAL  SURVEYING. 


Any  one  of  the  systems  of  coordinates  which  have  been  presented 
under  ' '  Horizontal  Surveying "  (Part  II)  can  be  used  in  Vertical 
Surveying,  but  for  practical  reasons  the  only  systems  commonly  em- 
ployed are  rectangular,  polar  and  polar-rectangular  coordinates. 

It  will  be  well  to  bear  in  mind  the  fact  that  while  only  one  hori- 
zontal plane  can  be  drawn  through  a  given  point,  an  infinite  number 
of  vertical  planes  can  be  passed  through  the  point. 

Nearly  all  vertical  surveying  is  for  the  purpose  of  determing  eleva- 
tions, in  which  case  it  is  called  leveling. 


CHAPTER   X. 
RECTANGULAR  COORDINATES. 

67.  Rectangular  Coordinates. —  In  Fig.  42  we  have  a  point  Pn 
referred  to  the  two  rectangular  coordinate  axes  Zk  and  Dk  (which 
intersect  at  the  origin  P^)  by  the  rectangular  coordinates  dk.n,  zk.n. 


>Pn 

Z 


Datum  Line 


K-n 


*+o* 


(Horizontal) 

FIG.  42  FIG.  43 

The  two  axes  lie  in  the  vertical  plane  determined  by  Pk  and  Pn. 
The  </-axis,  Z^, ,  is  a  hortizontal  line  and  is  called  datum  line. 
The  #-axis,  Zk ,  is  a  vertical  line.    dfc.n  is  the  (horizontal)  distance 
ofP.fromP,. 

^.;z  is  the  elevation  of  P;z  referred  to  datum. 


RECTANGULAR    COORDINATES 


53 


If  a  point  is  above  the  datum  line  its  elevation  is  considered 
positive;  if -below,  negative. 

The  shaded  line  Pl  P.2  P3 ,  Fig.  43,  represents  a  section  (called 
profile)  of  the  earth's  surface  cut  by  a  vertical  plane.  Dl  is  a  datum 
line  drawn  through  P1  and  in  the  plane  of  the  section.  Z^  is  a  verti- 
cal line  drawn  through  Pl ,  and  is  the  ^-axis. 

P2  is  #1>2  below  the  datum  line  and  horizontally  distant  drz  from 
P!  .  P3  is  zrz  above  the  datum  and  horizontally  distant  drz  from 
P!  ,  the  origin.  All  these  relations  are  expressed  briefly  by  writing 


68.  From  General  to  Local  Rectangular  Coordinates. —  Fig.  44. 
Given  the  general  coordinates  dk ,    zk  of  the  point  Pk ,  and  dn ,    zn  of 
the  point  PM  .    Required  the 
local  coordinates  d,,.,,,  #*.„  of     — 7  jJ7 

_l_    /  ~ /. y^ 

the  point  Pn  referred  to  the  •          -L __A 

CI  K  •  r\ ^1 

local  axes  /?A  and  Zk  drawn 

through  the  local  origin  Pk . 

Or,  briefly, 

Given     Pk  (dk,    ^,), 

and         Pn  (dn ,    ^), 

Required       Pw  (^.w ,    ^.w). 

Evidently    dk.w  =  dn  —  dfc. 


=  elevation  difference 


General   Datum 
FIG.  44 


where  the  signs  of  all  quantities  are  to  be  considered  when  making 
numerical  substitutions. 

69.  From  Local  to  General  Rectangular  Coordinates  ___  Fig.  44. 
Given  the  coordinates  dk.H  ,  zfc.n  of  the  point  Pn  referred  to  the  local 
origin  Pk  ,  and  the  coordinates  dk  ,  zk  of  Pk  referred  to  the  genarel 
origin.  Required  the  general  coordinates  dn,  zn  ofPn.  Or,  briefly, 
Given  P«(^.«,  ^-«)  and  P,  (^,  zk\ 

Required  Pn(dn,    ^). 

Evidently  dn  =  dk  +  dA.n  ,     ZH  =  zk  -f  ^  , 

in  which  the  signs  of  all  quantities  are  to  be  considered  when  making 
numerical  substitutions. 


54 


COORDINATES    OF    ELEMENTARY    SURVEYING 


70.  From  Local  Rectangular  to  Polar  Coordinates. —  Fig.  45. 
Given  the  rectangular  coordinates  dk.n ,  zk.n  of  the  point  Pn  referred 
to  the  pole  Pk .  Required  the  polar 
coordinates  <rk.n ,  sdk.n  of  Pn  referred 
to  datum  Dk  and  the  pole  Pk .  Or, 
briefly, 

Given  Pn  (d*.n,    ^«)> 

Required     Pn  (trk.H ,    sdk.t^). 
The  tangent  of  the  slope  is 

tan  <rk.n  =  zk.n  /  dk.n. 
The  slope-distance  is 

sdk.M  =  zk.n  I  sin  ov« . 
In  these  equations  the  signs  of  the  quantities  must  be  considered. 

The  check  equation  is 


71.  From  General  Rectangular  to  Polar  Coordinates. — Fig.  46. 

Given  the  rectangular  coordinates 
dk ,  zk  of  the  point  PA ,  and  dn ,  zn 
of  the  point  Pn ,  referred  to  the 
same  axes.  Required  the  polar 
coordinates  <rk.n ,  sdk.n  of  Pn ,  re- 
ferred to  datum  Dk  and  pole  Pk . 
Or,  briefly, 

P* 


Given 


k  ,    zk  ) 


FIG.  46 

is  tan  <rk.n  —  (zn  —  ZA 

The  slope  distance  is 

sdk.n  =  (zn  —  ZK)  I  sin 
=  A#£.M  /sin  (rk.n . 
The  check  equation  is 


and  Pn  (dn,    zn\ 

Required     Pn  (>>„,    J^..,,). 
The  tangent  of  the  slope  angle 


72.  From  Lccal  Rectangular  to  p 
Polar -Rectangular   Coordinates.- 
Fig.  47.     Given  the  rectangular  co- 


FIG.  47 


RECTANGULAR    COORDINATES  55 

ordinates  dA.Ht   zk.n  of  the  point  Pn  referred  to  the  poleP*.     Re- 
quired the  polar-rectangular  coordinates  <rk.n^    dk.n  of  Pn  referred  to 
datum  line  Dk  and  pole-origin  Pk .     Or,  briefly, 
Given  PM  (d»H ,    ^.M), 

Required  Pn  (<r*w,    </„„). 

The  tangent  of  the  slope  angle  is 

tan  <rk.,t  =  zk.n  I  d*H. 
The  (horizontal)  distance  is 


CHAPTER   XI. 


POLAR  COORDINATES. 

73.  Polar  Coordinates. —  In  Fig.  48  is  shown  a  point  /^referred 
to  the  line  Dk  and  the  pole  Pk  by  the  polar  coordinates  crA.H  and  sdk.n . 
ck.n  is  the  slope  angle,  and  sdk.n  is  the  slope  distance. 


(Horizontal) 
FIG.  48 


FIG.  49 


d.    — 


cos  arA.n. 


74-  From  Polar  to  Local  Rectangular  Coordinates. —  Fig.  49. 
Given  the  polar  coordinates  ork.H ,    sdk.n  of  the  point  Pn  referred  to 
the  datum  line  Dk  and  the  pole  Pk .    Required  the  rectangular  co- 
ordinates dA.nt    zk.n  of  PH  referred  to  the  origin  Pk.     Or,  briefly, 
Given  Pn  (crk.n  ,    s^,.n), 

Required  Pn  (dk.n,    ^.M). 

The  (horizontal)  distance  is 
The  elevation  is 

zk.n  =--  sdk.n  sin  (rk.n  ; 
and  the  check  equation  is 

75  ^  From  Polar  to  Polar- 
Rectangular  Coordinates.  —  Fig. 
50.  Given  the  polar  coordinates 
<rk.n ,  sdk.n  of  the  point  Pn  referred 
to  the  datum  line  Dk  and  the  pole 


POLAR    COORDINATES  57 

PA.     Required  the  polar-rectangular  coordinates  ak.n>   dk.n  of  PM 
referred  to  the  datum  line  Dk  and  the  pole-origin  Pk  .     Or,  briefly, 


Required  Pn  (crk.H  ,    ^.M). 

The  slope  angle  is 

0V«  =s=  o-A.M. 
The  horizontal  distance  is 

dk.n  =  j^.M  cos  o-A.H. 


CHAPTER   XII. 
POLAR-RECTANGULAR  COORDINATES. 

76.  Polar-Rectangular  Coordinates — In  Fig.  51  is  shown  a  point 
Pn  referred  to  the  datum  line  Dk  and  pole-origin  Pk  by  the  polar- 


rectangular  coordinates  ak.n ,    dk 
the  (horizontal)  distance. 


is  the  slope  angle;  dk.n  is 


77.  From  Polar-Rectangular  to  Local  Rectangular  Coordinates. 

Fig.  52.  Given  the  polar-rectangular  coordinates  <rk.n,  dk.n  of 
the  point  Pn  referred  to  the  datum  line  Dfc  and  the  pole-origin  Pk . 
Required  the  rectangular  coordinates  dk.n ,  zk.n  of  PH  referred  to  the 
origin  Pk.  Or,  briefly, 

Required  /VwT,'  O-' 

The  (horizontal)  distance  is 


The  elevation  is 

zk.n  =  dk.n  tan  <rk.n . 

78.  From  Polar-Rectangular 
to  Polar  Coordinates.  —  Fig.  53. 
Given  the  polar-rectangular  coord- 
inates ov«>  dfrn  °f  the  point  Pn 
referred  to  the  datum  line  Dk  and 


POLAR-RECTANGULAR    COORDINATES  59 

the  pole-origin  P \ .     Required  the  polar  coordinates  crk.H ,  sdk.n  of  Pn 
referred  to  the  datum  line  Dk  and  the  pole  Pk .     Or,  briefly, 

Given  Pn  (ak.n ,    */*.„), 

Required  PH  (0*.M,    J^.M). 

The  slope  angle  is 

<r*-n  *=  «•*•«• 
The  slope  distance  is 

sdk.n  =  dt.H  I  CQS  <rA... 


PART  IV.— SPACE  SURVEYING. 

In  Horizontal  Surveying  (Part  II)  we  considered  only  the  hori- 
zontal relations  between  points  :  i.  e. ,  only  the  relations  between  pro- 
jections of  points  on  a  horizontal  plane.  In  Vertical  Surveying  we 
considered  only  the  relations  between  points  lying  in  a  vertical  plane. 
In  Space  Surveying  we  have  a  combination  of  Horizontal  and  Verti- 
cal Surveying,  and  consider  the  space  relations  between  points. 


CHAPTER    XIII. 
RECTANGULAR  COORDINATES. 

79.  Rectangular  Coordinates.  —  Fig.  54.  The  datum  plane  is 
the  horizontal  plane  passed  through  the  origin  Pk  .  The  axes  X  k 
and  Yk  intersect  at  right  angles  at  Pk  .  From  any  point  Pn  in  the 

datum  plane,  draw  the 
JL#*.J  to  Yk,  andj^ 
to  Xk.  x»nt  yk.n  are 
the  rectangular  coordi- 
nates of  Pn  referred  to 
the  axes  Xk  and  Y  k. 

Let  Pn  be  a  point 
at  a  distance  zk.n  verti- 
cally above  PH'.  Pn'  is 
the  projection  of  PHon 
the  datum  plane.  Evi- 
dently the  position  of 
Pn  with  respect  to  the 
axes,  Xfc  and  Ykt  and 
the  datum  plane,  is  de- 
termined by  the  three 

quantities  xk.n,  yk.n,  zk.n.  "Pn(xk.n,  yk.n,  £*.„)"  means  "/^isa 
point  situated  at  a  distance  sk.n  vertically  above  a  point  (of  the  datum 
plane)  whose  rectangular  coordinates  arejr^,  yfc.n." 


pIG 


RECTANGULAR    COORDINATES 


61 


If  we  are  considering  only  the  elevations  of  points  of  a  survey, 
we  ignore  them's  and  jj/'s  and  take  account  of  the  #'s  only.  On 
the  other  hand  when,  in  any  part  of  a  problem,  we  have  to  con- 
sider horizontal  relations  only,  we  use  the  ;tr's  and  j/'s  and  ignore 
the  z's. 

80.  From  General  to  Local  Rectangular  Coordinates. — Fig.  55. 
Given  the  rectan- 
gular coordinates 

%k  >  y& »  %k  °f the 

point  Pj,  and^TM, 
yn,  zn  of  the  point 
Pn  referred  to  the 
same  origin.  Re- 
quired the  rectan- 
gular coordinates 

PM  referred  to  the 
origin  Pk ,  the  lo- 
cal axes  Xk ,  Yk , 
Zk  being  taken 
parallel  respect- 
ively to  the  gen- 
eral axes  X,  Y,  Z. 
Or,  briefly,  Given 
Required 


General     Datum 


FIG.  55 


Pk  (xk ,  yk , 
Pn\ 

An  inspection  of  Fig.  55  shows  that 


and 


The  signs  of  the  coordinates  must  be  considered  when  substitut- 
ing numerical  values. 

81.  From  Local  to  General  Rectangular  Coordinates.  Single 
Point. —  Fig.  55.  Given  the  coordinates  xky  yky  zk  of  the  point 
Pk  referred  to  a  general  origin  ;  and  the  coordinates  xk.n,  yk.n ,  zk.n 
of  the  point  Pn  referred  to  the  local  origin  Pk ,  the  general  axes  being 
parallel  respectively  to  the  corresponding  local  axes.  Required  the 


corrdinates  xn 
briefly, 


yn ,    zn  of  Pn  referred  to  the  general  origin.     Or, 


62 


COORDINATES    OF    ELEMENTARY    SURVEYING 

*fo.«»     y*n>      Zk-n 


Given  Pk(xk,  y**    **)     and     PH(x 

Required  /*„(#„,  y,t,    z,t). 

An  inspection  of  Fig.  55  shows  that 

Xn  r==  %k   ~T  %k'n) 


and 


z   = 


The  signs  of  the  coordinates  must  be  considered  when  making 
numerical  substitutions  in  these  equations. 

It  is  evident  that  the  x's  and  jy's  can  be  computed  as  in  Art.  21, 
and  the  z's  computed  by  themselves  ;  or  the  three  coordinates  can 
be  computed  in  one  algorithm. 

82.  From  Local  to  General  Rectangular  Coordinates.  Series 
of  Points.  —  Given  a  series  of  points,  Plt  P^,  .  .  .  .  Given  the  gen- 
eral coordinates  x^  ,  yl  ,  zl  ,  of  the  first  point  Pl  of  the  series.  For 
each  point  after  the  first  we  are  given  the  local  coordinates  referred 
to  the  preceding  point  as  origin.  All  the  ^r-axes  are  parallel  ;  all 
the  j-axes  are  parallel  ;  and  all  the  ^-axes  are  parallel.  Required 
the  general  coordinates  of  any  point  of  the  series.  Or,  briefly, 

Given  Pl  (X  ,   j,  ,    ^), 

Pi(*r,>   -Ti-2>    *i-.), 

*    3  V^-3  »      J^2'3  »       ^2-S  >> 

Required  Pn(Xn*  ?„,    ^«). 

The  required  coordinates  of  PH  are 

xn  =  x^  -f  xvi  +  XY*  +  •   .   .  +  xn_rn, 

y*  =}'i  -\-yr-*  +72-3  -f  •  .  .  -f-  j*w«, 
^«  =  2,  +  ^.,  +  *a.8  +  .  .  .  -f  rw_1>w. 

Or,  in  words, 

abscissa 


The  general 


ordinate 
elevation 


of  any  point  of  the  series  is  equal  to  the 


algebraic  sum  of  the  local 


abscissas 
ordinates 
elevations 


of  all  the  points  after  the  first 


point  up  to  and  including  the  given  point,  and  the  general 
of  the  first  point. 


abscissa 
ordinate 
elevation 


RECTANGULAR    COORDINATES 


If  we  are  considering  the  lines  Pl 
coordinates  may  be  expressed  thus  : 


63 
.  the  required 


ytt  = 

Or,  in  words : 
The  general 


.2  -f 


a.s  -f  .  .  . 


abscissa 
ordinate 
elevation 


of  any  given  point  of  the  series  is  equal 


to  the  algebraic  sum  of  the  general 


I  x- 


abscissa 
ordinate 
elevation 


of  the  first  point  and 


the 


differences  of  all  the  lines  between  the  first  point  and  the 


given  point. 

Them's  andjy's  can  be  considered  by  themselves  (as  in  Art.  22) 
and  the  ^'s  separately  considered  (as  in  Art.  69);  or  the  three  co- 
ordinates can  be  computed  in  one  algorithm. 

83.  From  General  Rectangular  to  Polar  Coordinates. — Fig.  56. 
Given  the  rectangular 
coordinates  xk>  yk,  zk 
of  the  point  Pk ,  and  xn  , 
yn ,  zn  of  the  point  Pn 
referred  to  the  same 
origin.  Required  the 
polar  coordinates  aA.M, 
<rA.M,  sdk.n  (i.  *.,  azi- 
muth, slope  angle  and 
distance).  Zero-azi- 
muth is  -\-Yjt.  (Tk.n  is 
referred  to  datum  plane 
through  Pk ,  and  Pk  is 
the  pole.  Or,  briefly, 


Required  Pn(ak.n, 

Fig.  56  shows  that  the  tangent  of  the  azimuth  is 
tan  ak,n  =  (xn  —  *,)  /  (yn  —  yk)  =  A^.w 
the  tangent  of  the  slope  angle  is 


64 


COORDINATES    OF    ELEMENTARY    SURVEYING 

I  sin  a,.,,  = 


tan  vk.n  =  [(X,  —  Zt)  I  (xn  — . 
and  the  slope  distance  is 

sdk.n  =  (zn  —  z^  I  sin  afc  n  = 
The  check  equation  is 


sn 


sn 


84.  From  General  Rectangular  to  Polar-Rectangular  Coordi- 
nates.—  Fig.  57.  Given  rectangular  coordinates  xk*  yk,  zk  of  the 
point  Pk ,  and  xn  ,  yn ,  zn  of  the  point  Pn  referred  to  the  same  ori- 
gin. Required  the  polar-rectangular  coordinates  ak.n ,  at.nt  dk.HQ\ 
Pn  referred  to  the  zero-azimuth  plane  Zk  Yk ,  the  datum  plane  DA 

and   the   pole-origin  Pfc. 
y  7  \  n  Or,  briefly, 

~*      ^  "  Given      Pk(xk,  yk,    z,) 

and     Pn(xnt  yn,    z,t), 
Required 


y  The  tangent  of  the  azi- 

^     muth  is 


The  (horizontal)  distance 
-X     is 

FIG.  57  dk-n  =  (*n  —  x^  /  sin  ak.n 

=  kxk.n  I  sin  ak.n  . 

Tangent  of  slope  angle  is    tan  <rk.n  =  (zn  —  zk)  /  dk.n  =  Azk.n  /  dk.n  . 
A  check  equation  is 


CHAPTER  XIV. 
POLAR  COORDINATES. 

85.  Polar  Coordinates. —  In  Fig.  58  Xk  and  Yk  lie  in  a  horizon- 
tal plane  containing  the  point  P^ .     Zk  is  a  vertical  line  containing 
Pk.     Pk  Pn  P^  lies  in  the  vertical  plane  determined  by  Zk  and  Pn. 
ak.n  is  the  angle  between  the  two  vertical  planes  Zk  Yk  and  Zfc  Pn. 
crk.n  is  the  angle  which  the  line  Pk  Pn  makes  with  the  horizontal  plane 

The  point  Pn  is  referred  to  — 
the  zero-azimuth  plane,  ZkYk) 

the  datum  plane  Dk  (i.  e.,  the  horizontal  plane  through  Pk*)t 
and  the  pole  Pk , 
by  the  polar  coordinates  — 
ak.n ,  called  azimuth, 
<rk.n ,  called  slope  angle,  and 
sdk.n<  called  slope  distance. 

86.  From  Polar  to  Polar-Rectangular  Coordinates.— Fig.  58. 
Given  the  polar  coordinates  ak.n ,    <rk.n ,    sdh.n ,  of  the  point  Pn  re- 
ferred to  the  pole  Pk.     Required  the 

polar  rectangular  coordinates  ak.n,  <rk.n* 
dk.n  of  Pn  referred  to  the  same  zero-azi- 
muth plane,  the  same  datum  plane  Dk 
and  to  the  pole-origin  Pk.     Or,  briefly, 
Given         PH(afc.nf    <rfc.n,    sdk.n\ 
Required         Pn(^-n,    0W»    d*n)> 
Evidently,          ak.n  =  ak.n 


and         4>w  =  ^.^  .  cos 

87.  From  Polar  to  Local  Rectangu-  FlG>  58 

lar  Coordinates. —  Given  the  polar  coordinates  afc.n,  (rk.ni  sdk.n  of 
the  point  Pn  referred  to  the  pole  Pk.  Required  the  rectangular 
coordinates  xk.n ,  yk.n ,  zk.n  of  Pn  referred  to  the  origin  Pk ;  the 
axes  being  so  taken  that  Zk  and  Yk  lie  in  the  zero-azimuth  plane  of 
polar  coordinates.  Or,  briefly, 


66  COORDINATES    OF    ELEMENTARY    SURVEYING 

Given  Pn(ak-n>    o-k.nt    sdk.n^ 

Required  Pn  (xk.n ,  yk.n ,    zk.n ). 

In  Fig.  57  dk.n  =  sdk.n  cos  <rk.n , 

and  the  required  coordinates  are 

*k-n  =  dk-n  sin  ak-n> 

V*.*.  =  d}t.n  cos  al 


k.n 


k.n  , 


The  check  equation  is        x\.n  +  y\.n  -f  z\.n  =  (sdk.n)\ 

Example.  —  sdr2  =-  20,  sdr3  =  40,  aV2  =  340°,  ar3  =  110°, 
o-r2=  12°,  o-1>3=  —8°.  Required  ^.2,  7^,  ^.2  ,  and  ^r3,  ^.3, 
^1-3  ;  +  Y  (axis)  being  taken  as  zero  azimuth. 


I 

/\/> 

P.P. 

2 

a 

34°o 

110° 

3 

(T 

12° 

—8° 

4 

sd 

20. 

40. 

5 

log  sd 

I.30I 

i  .602 

6 

log  cos  cr 

9.990 

9.996 

7 

log^ 

I.29I 

1.598 

8 

log  sin  a 

9-534" 

9-973 

9 

log  cos  a 

9-973 

9-534n 

10 

log  sin  cr 

9.318 

9-144" 

ii 

log  x 

0.82  ^n 

1  .  571 

12 

log/ 

1.264 

i.i32n 

13 

log* 

0.619 

o.746n 

14 

X 

-6.7 

37-2 

15 

y 

18.4 

-13-6 

16 

4.2 

-5-6 

17 

log;r2 

1.650 

3-I42 

18 

log  yj 

2.528 

2  .  264 

19 

log  £T2 

1.238 

1.492 

20 

log  sd2 

2  .6O2 

3.204 

21 

X* 

44-7 

1387. 

22 

j/2 

337- 

184. 

23 

^2 

17  3 

31- 

24 

*•+/.+  22 

399- 

1602. 

25 

check  :   (j<^)2 

400. 

1600. 

88.  From  Polar  to  General  Rectangular  Coordinates. — Change 
(i)  from  polar  to  local  rectangular  (Art.  87)  and  (2)  from  local  rectan- 
gular to  general  rectangular  coordinates  (Art.  81  or  Art.  82). 


CHAPTER  XV. 


POLAR-RECTANGULAR  COORDINATES. 

89.  Polar-Rectangular  Coordinates. — The  polar  coordinates  be- 
come the  polar-rectangular  coordinates  of  a  point  if  we  substitute 
horizontal  distance  for  slope  distance. 

In  Fig.  59  the  point  Pn  is  referred  to  — 
the  zero-azimuth  plane  Z^  Yk , 

the  datum  plane  Dk  (i.  e.,  the  horizontal  plane  J£kYk")t  and 
the  pole-origin  Pk , 

by  the  polar-rectangular  coordinates  — 
ak.n ,  called  azimuth, 
<rfc.n ,  called  slope  angle,  and 
dk.n  called  (horizontal)  distance. 

90.  From  Polar-Rectangular  to  Local  Rectangular  Coordinates. 
Given   the   polar-rectangular   coordinates    ak.n,    trk.nt    dk.n  of  the 
point  Pn  referred  to  the  pole-origin  Pk .     Required  the  rectangular 
coordinates  xk.n,  yk.ny  zk.n  of  pre- 
ferred to  the  origin  Pk ,  the  axes  Zk , 

Yk  lying  in  the  zero-azimuth  plane 
of  the  polar-rectangular  coordinates. 
Or,  briefly, 

Given  />n («*.„,    <rfc.»>    dk.n), 

Required     Pn(xk.n)  yk.n,    zk.n). 

An  inspection  of  Fig.  59  will  show 
that         x^n  =  dk>n  sin  ak.n 

yk-n  =  4-n  cos  ak-n 

zk-n  =  dk-n  tan  o-k.n . 


FIG.  59 


91.  From  Polar-Rectangular  to  General  Rectangular  Coordi- 
nates.— After  finding  the  local  coordinates,  convert  these  into  gen- 
eral coordinates  by  Art.  81  or  Art.  82. 

92.  From  Polar-Rectangular  to  Polar  Coordinates. —  Given  the 
polar-rectangular  coordinates  ak.n,    <rk.nt    dk.n  of  the  point  Pn  re- 


68  COORDINATES    OF    ELEMENTARY    SURVEYING 

ferred  to  the  pole-origin  Pk.  Required  the  polar  coordinates  afc.n, 
°Vn  i  sdk-n  ofPn  referred  to  the  pole  Pk  ;  the  zero-azimuth  and  datum 
planes  being  the  same  for  both  systems  of  coordinates.  Or,  briefly, 

Given  Pn(ak-n>     OVn»     ^4-n), 

Required  Pn(ak-n>    0Vn>    -"4-n)- 

Evidently,  ak.n  =  ak.n, 


and 


=  dk.n  /  cos  ak.n  . 


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